ijf^a 


f 


IN   MEMORIAM 
FLORIAN  CAJORl 


/-u 


it^try 


TREATISE  ON  ALGEBRA, 


EMBRACING, 


BESIDES  THE  ELEMENTARY  PRINCIPLES,  ALL  THE 
HIGHER  PARTS  USUALLY  TAUGHT  IN 


COLLEGES 


COXTAINING 


MOREOVER,  THE  NEW  METHOD  OF  CUBIC  AND    HIGHER  EQUATIONS 

AS  WELL  AS  THE  DEVELOPMENT  AND  APPLICATION 

OF  THE  MORE  RECENTLY  DISCOVERED 


THEOREIM  OF  STURM. 


BY  GEORGE  R.  PERKINS,  A.  M., 

PROFESSOR   OP   MATHEMATICS    IX    NEW-VORK   STATE    NORMAL  SCHOOL.   AUTHOR 
"  ELEMENTART   ARITHMETIC,"'    '"  HIGHKR    ARITHMETIC,"    "'  ELEMENTS 
OF   ALGEBRA,"    ETC.,    ETC. 


SECOND    EDITION,    REVISED,    ENLARGED    AND    IJIPROVED. 

NEW    YORK: 
PUBLISHED  BY  D.  APPLETON  AND  CO. 

UTICA: 

HAWLEY,  FULLER  AND  CO. 

1850. 


FIRST   EDITION. 

Entered,  according  to   Act   of   Congress,  in   the   year   1841,  by 

GEORGE  R.  PERKINS, 
in  the  Clerk's  Office  of  the  Northern  District  of  New- York. 


SECOND  EDITION. 
Entered,  according  to  Act   of   Congress,  in  the  year  1847,  by 

GEORGE  R.  PERKINS, 
in  the  Clerk's  Office  of  the  Northern  District  of  New-York. 


C.    VAN    BENTHUYSEN    AND    CO. 
fiTEREOTYPERS. 


PREFACE 

TO    THE    FIRST    EDITION. 


In  presenting:  this  volume  to  the  Public,  I  would  not  claim  to 
have  unfolded  many  new  principles  of  Algkgra;  I  only  claim  to 
have  judiciously  combined  and  arraii^^ed  principles  already  known. 
By  commencing  this  work  with  the  most  elementary  parts,  and 
gradually  ascendinjj  to  the  more  comftlicated,  I  have  designed  to 
adapt  it  to  the  wants  of  students  of  every  grade. 

While  I  acknowledge,  that,  in  general,  the  principles  have  long 
been  known,  I  think  I  am  justifiable  in  claiming  some  of  the  methods 
of  operation  as  original. 

This  work  will  be  found  to  contain,  for  the  first  time,  I  believe 
in  any  American  school  book,  a  demonstration  and  application  of 
Stubm's  Theohkm;  by  the  aid  of  which,  we  may  at  once  deter- 
mine the  number  of  real  roots,  of  any  Algebraic  Equation,  with 
much  more  ease,  than  could  be  done  by  any  previously  discovered 
methods. 

The  method  of  finding  the  numerical  values  of  the  roots  of 
cubic  and  higlier  equations,  as  fully  explained  under  the  last  chapter, 
will,  no  doubt,  be  new  to  many,  and  interesting  to  all  lovers  of  this 
science.  It  is  particularly  interesting  on  account  of  the  ease  witli 
which  it  resolves  itself  into  the  method  of  extracting  any  root  of  a 
number,  as  explained  in  my  IIioiikk  Akitiimetic. 

It  would  be  extrcmelv  diflficult  to  point  out  the  exact  sources 
from  which  I  hiive  drawn  for  this  work,  and  even  could  I  do  so,  these 
principles  have  been  so  long  in  use,  we  could  not  with  safety  say 
when,  and  with  whom,  they  each  originated.  While  I  acknowledge 
the  aid  of  many  works  on  this  science,  I  would  give  by  far  the 
greatest  share  of  credit,  to  the  eighth  edition  of  Bourdon's  most 
excellent  treatise  on  Algebra. 

rtica,  July,  1842.  GEORGE  R.  PERKINS. 


iv'tmn-9i'<a 


PREFACE 

TO    THE   SECOND    EDITION. 


TiiK  ]>ri'sei.t  Edition  luis  been  very  carefully  Revised  and 
cinsideiably  Enlarg'cd.  One  entire  Chapter  on  tiic  subject  of  Con- 
•ii.xuED  FkactU/Xs,  which  are  treated  in  quite  a  general  manner 
has  been  added.  The  subject  of  Recurring  Series  has  been 
re-written,  and  n-.uch  ^^n1pli^led,  and  many  other  chans^es,  which 
we  deemed  to  be  ini'provemenls,  have  been  introduced. 

Ilavin;^  almost  daily  made  lue  of  tl.is  w<^'rk  in  my  Classes,  since 
its  Publication,  and  always  having-  had  in  view  tho  ciianges  which  it 
would  be  desirable  to  make,  in  order  to  improve  the  work,  we  feel 
that  we  are  now  ])rej  arcil  to  present  thi*  pr^  sent  edition  as  quite  an 
improvement  upon  ihe  first.  It  is  bclitved  il  will  be  found  to  con- 
tain a  i)rclty  lull  ami  C(  nijilele  dcvelopmrnt  of  all  ihe  \arioussul) 
jects  of  Algebra,  usually  taught  in  our  Colleges. 

As  wehavealrcady  ])rej);ire(l  a  smalhr  work,  especially  dcsignod 
tor  i)rimary  schools,  it  has  been  our  aim  to  adapt  this  Treatise  to 
the  wants  of  the  more  advanced  Schools  and  Colleges. 

Utica,  Januarti,  1S17.  GEORGE  R.  PE.'vKI.'.'S. 


CONTENTS. 


CHAPTER  I. 

DEFINITIONS    AND    PRELIMINAKY    RUI.Ei>. 

Definitions, 9 

Addition, 15 

Subtraction, 21 

Multiplication, 24 

Division 30 

CHAPTER  11. 

ALGEBRAIC    FRACTIONS. 

Reduction  of  monomial  fractions. 38 

Greatest  common  measure , 40 

Reduction  of  polynomial  fractions, 46 

Redaction  of  a  mixed  quantity  to  the  form  of  a  fraction, 48 

Reduction  of  a  fraction  to  an  entire  or  mixed  quantity, 50 

Reduction  of  fractions  to  a  common  denominator, 52 

Addition  of  fractions, 53 

Subtraction  of  fractions, 54 

Multiplication  of  fractions, 56 

Division  of  fractions, 57 

CHAPTER  III. 

SIMPLE    EQUATIONS. 

Equations  defined, 59 

Axioms  used  in  solving  equations CO 

Clearing  equations  of  fractions, 60 

Transposinij  the  terms  of  an  equation C,2 

General  rule  for  solving  simple  equations 64 

QuPitions  involving  simple  equations 67 

ECiUATIONS    OF    TWO    OR    MORE    UNKNOWN    QUANTniES. 

Elimination  by  addition  and  subtraction, 79 

Elimination  by  comparison, 84 

Elimination  by  substitution, 87 

Questions  involving  two  or  more  unknown  quantities, 91 

General  solution  of  literal  equations, 106 

Checker-board  process, 113 


n 


CHAPTER  IV. 


INVOLUTION. 

To  involve  a  monomial, 122 

To  involve  a  polynomial, 124 

EVOLUTION. 

To  extract  a  root  of  a  monomial, 127 

General  rule  for  extracting  any  root  of  a  polynomial, 131 

Enunciation  of  the  square  of  any  polynomial, 133 

Rule  for  extracting  the  square  root  of  a  polynomial, 134 

New  rule  for  extracting  the  cube  root  of  a  polynomial, 136 

SURD    QUANTITIES. 

To  reduce  a  quantity  to  the  form  of  a  surd, 141 

To  reduce  surds  to  a  common  index, 142 

To  reduce  surds  to  their  simplest  form, 143 

Addition  and  subtraction  of  surds 145 

Multiplication  and  division  of  surds, 146 

Extraction  of  the  square  root  of  a  binomial  surd, 147 

To  find  multipliers  which  will  cause  surds  to  become  rational,....  152 

IMAGINARY    QUANTITIES. 

Defined, 159 

Multiplication  of  imaginaries, 161 

Division  of  imaginaries, 162 

Interpretation  of  the  symbols --,  — ,   -, io6 

CHAPTER  V. 

QUADRATIC    EQUATIONS. 

Incomplete  quadratic  equations, 168 

Complete  quadratic  equations, 172 

First  rule  for  quadratic  equations, 174 

Second  rule  for  quadratic  equations, 180 

Equations  of  several  unknown  quantities  involving  quadratic  equa- 
tions,   181 

Questions  involving  quadratic  equations 196 

Properties  of  the  roots  of  quadratic  equations, 202 

Examples  giving  imaginary  roots, 214 

CHAPTER  VI. 

RATIO    AND    PROGRESSION. 

Ratio  defined, 217 

Table  giving  formulas  for  the  20  cases  of  arithmetical  progression,  221 

Geometrical  ratio, 224 

Table  of  all  the  formulas  of  geometrical  progression 230 

Harmonical  proportion, 234 


CONTENTS,  VU 

CHAPTER  VII. 

Method  of  indeterminate  coefficients, 237 

Binomial  Theorem  demonstrated, 243 

Application  of  the  binomial  theorem, 248 

Multinomial  Theorem  demonstrated, 253 

Examples  under  the  multinomial  theorem, 257 

Reversion  of  series, 258 

Differential  method  of  series, 261 

Summation  of  infinite  series, 267 

Recurring  series, 272 

A  geometrical  series  may  be  also  a  recurring  series, 277 

CHAPTER  VIII. 

COMTINUED    FRACTIONS. 

Defined, 283 

General  rule  for  converting  a  continued  fraction  into  its  approxima- 
tive values, 286 

To  convert  a  common  fraction  into  a  continued  one, 287 

Rule  for  converting  the  common  kind  of  continued  fractions  into  its 

approximative  ralues, 294 

The  square  roots  of  surds  found  by  continued  fractions, 295 

CHAPTER  IX. 

LOGARITHMS. 

Defined, 309 

Logarithmic  formula  found 311 

Numerical  calculation  of  logarithms, 313 

Exponential  Theorem 320 

Application  of  logarithms, 323 

Exponential  equations  resolved  by  logarithms, 325 

Compound  interest  and  annuities  by  logarithms 326 

CHAPTER  X. 

OEWERAL    PROPERTIES    OF    EqUATIONS. 

The  root  of  an  equation  defined 336 

The  number  of  positive  and  negative  roots, 339 

The  relation  between  the  roots  and  coefficients, 341 

Impossible  roots  occur  in  pairs, 342 

The  sum  of  the  7nth  powers  of  the  roots, 34-1 

To  cause  the  second  term  to  vanish, 345 

Method  of  finding  the  derived  polynomials, 348 

Equations  having  equal  roots, 352 

Recurring  equations, 355 


Vlll  CONTENTS. 

Binomial  equations, 359 

General  solution  of  an  equation  of  the  third  degree, 365 

General  solution  of  an  equation  of  the  fourth  degree. 372 

Sturm's  Theorem  defined, 375 

Demonstration  of  Sturm's  theorem, 376 

Application  of  Sturm's  theorem, 380 

General  method  of  elimination  among   equations   above  the  first 
degree, 385 

CHAPTER  XI. 

Numerical  solution  of  cubic  equations, 392 

Numerical  solution  of  equations  above  the  third  degree 413 


I 


TREATISE  ON  AL&EBRA. 


CHAPTER  I. 

DEFINITIONS  AND  PRELIMINARY  RULES. 

DEFINITIONS. 

{Article  1.)  Algebra  is  that  branch  of  Mathematics,  in 
which  the  calculations  are  performed  by  means  of  letters  and 
signs  or  symlols. 

(2.)  In  Algebra,  quantities,  whether  given  or  required, 
are  usually  represented  by  letters.  The  first  letters  of  the 
alphabet  are,  for  the  most  part,  used  to  represent  known 
quantities ;  and  the  final  letters  are  used  for  the  unknown 
quantities. 

(3.)  The  symbol  =,  is  called  the  sign  oi  Equality  ;  and 
denotes  that  the  quantities  between  which  it  is  placed,  are 
equal  or  equivalent  to  each  other.  Thus  $1  =  100  cents, 
which  is  read,  one  dollar  equals  one  hundred  cents.  Again, 
a  =  6,  which  is  read,  a  equals  h. 

(4.)  The  symbol  -f?  is  called  plus  ;  and  denotes  that  the 
quantities  between  which  it  is  placed,  are  to  be  added 
together.  Thus,  a  -|-  6  =  c,  which  is  read,  a  and  h  added, 
equals  c.  Again,  a-\-h-\-c=d-\-x^  which  is  read,  o, 
h  and  c  added,  equals  d  added  to  x. 
2 


10  DEFINITIONS. 

(5.)  The  symbol — ,  is  called  minus;  and  denotes  that 
the  quantity  \vhich  is  placed  at  the  right  of  it  is  to  be  sub- 
tracted from  the  quantity  on  the  left.  Thus,  a  —  6  =  c, 
which  is  read,  a  diminished  by  b  equals  c. 

(6.)  The  symbol  X,  is  called  the  sign  oi  inultiplication ; 
and  denotes  that  the  quantities  between  which  it  is  placed 
are  to  be  multiplied  together.  Thus,  a  X  b  =  c,  which  is 
read,  a  multiplied  by  b,  equals  c.  Multiplication  is  also 
represented  by  placing  a  dot  between  the  factors,  or  terms 
to  be  multiplied.  Thus,  a  .  bis  the  same  as  a  X  b.  Another 
method,  which  is  used  as  frequently  as  either  of  the  above, 
is  to  unite  the  quantities  in  the  form  of  a  word.  Thus,  a6c 
is  the  same  as  a  X  6  X  c,  or  a  .  6  .  c. 

(7.)  The  symbol  -r,  is  called  the  sign  of  divisioji  ;  and 
denotes  that  the  quantity  on  the  left  of  it  is  to  be  divided 
by  the  quantity  on  the  right.  Thus,  a  -^  b  =  c^  which  is 
read,  a  divided  by  b  equals  c.  Division  is  also  indicated  by 
placing  the  divisor  under  the  dividend,  with  a   horizontal 

X 

line  between  them  like  a  vulgar  fraction.     Thus,  -  is  the 

fa  Jy 

same  as  x  -r  y. 

(8.)  When  quantities  are  enclosed  in  a  parenthesis,  brace, 
or  bracket,  they  are  to  be  treated  as  a  simple  quantity.  Thus, 
(a  4-  6)  -7-  c,  indicates  that  the  sum  of  a  and  b  is  to  be  divided 
by  c.  Again,  (x  — y)  -^z=[x  —  y]^z  =  \x  —  y]^Zj 
each  of  which  expressions  is  read,  y  subtracted  from  x  and 
the  remainder  divided  by  z.  The  same  thing  may  also  be 
expressed  by  a  bar  or  vinculum.  Thus,  x  —  y  -r-  z,  which 
is  read  the  same  as  the  last  three  expressions. 

(9.)  The  symbol  >,  is  called  the  sign  of  tne^wa/zYy;  and 
is  used  to  express  that  the  quantities  between  which  it  is 
placed  are  unequal.  Thus,  6  >  a  indicates  that  b  is  greater 
than  a  ;  and  2>  <  c  denotes  that  b  is  less  than  c. 


UEFINlTIOiVS.  11 

(10.)  When  a  quantity  is  added  to  ilscli  several  liiues,  as 
c  -\-  c  -'r  c  -{-  c,  we  need  write  it  but  onre,  by  placing  before 
it  a  number  to  show  how  many  times  it  has  been  taken. 
Thus,  f  -{-  c  -f-  c  -}-  c  =  4c.  The  number  which  is  thus 
placed  before  the  quantity  is  called  the  coefficitnt  of  the 
quantity.  In  the  above  example,  4  is  the  coefficient  of  c. 
A  coefficient  nniy  consist,  itself,  of  a  letter.  Thus,  n  is  the 
coefficient  of  x  in  the  ex])ression  nx;  so  also  may  x  be 
regarded  as  the  coefficient  of  n  in  the  same  expression. 

(11.)  The  continued  product  of  a  quantity  into  itself  is, 
usually,  denoted  by  writing  the  quantity  once,  and  placing 
a  number  over  the  quantity,  a  little  to  the  right.  Thus, 
o  X  (t  X  a  is  the  same  as  u^.  The  number  thus  placed  over 
the  quantity,  is  called  the  exponmt  of  the  quantity.  Thus, 
5  is  the  exponent  of  a  in  the  expression  a^,  and  denotes 
that  a  is  to  be  multiplied  into  itself,  as  a  factor,  five  times. 

(12.)  When  a  quantity  is  multiplied  continually  into 
itself,  the  result  is  called  a  power  of  the  quantity.  Thus,  a^ 
is  the  sixth  power  of  a,  and  a^  is  the  third  power  of  a,  the 
exponent  always  indicating  the  degree  of  the  power. 

W^hen  a  quantity  is  written  without  any  exponent,  it  is 
understood  that  its  exponent  is  a  unit. 

Thus,  a  is  the  same  as  a*,  and  (x  +  v)  X  m  is  the  same 
as  (x-f-y)'  X  in\ 

(13.)  The  symbol  v/,  is  called  the  radical  sign;  and' 
denotes  that  a  root  of  the  quantity,  over  which  it  is  placed,, 
is  to  be  extracted.     Thus, 

V  ar,  or  simply  -/  a:,  denotes  the  square  root  of  x. 

V  X  denotes  the  cube  root  of  x. 

V  X  denotes  the  fourth  root  of  x. 

The  number  placed  over  the  radical  is  called  the  index 
of  the  root.  Thus,  2,  3,  and  4  are,  respectively,  the  indices 
of  the  square  root,  cube  root,  and  fourth  root. 


12  DEFINITIONS. 

(14.)  A  root  of  a  quanlUy  may  also  be  represented  by 
means  of  a  fractional  exponent.     Thus,  the  square  root  of  a 

is  a";  the  cube  root  of  a  is  a';  th  •  fourth  root  of  a  is  a*  ; 
and  so  on  for  other  roots. 

By  the  same  notation,  a^  is  the  cube  root  of  the  square 
of  a,  or  the  square  of  the  cube  root  of  a.     For  the  same 

3 

reason  a^  is  the  fifth  root  of  the  third  power  of  «,  or  the 
third  power  of  the  fifth  root  of  a. 

(15.)  The  reciprocal  of  a  quantity  is  a  unit  divided  by 

that  quantity.  Thus,  -  is  the  reciprocal  of  a;  also  x  is  the 
reciprocal  of  3. 

(16.)  The  symbol  .-.,  is  equivalent  to  the  phrase,  there- 
fore, or  consequently. 

(17.)  When  algebraic  quantities  are  written  without  any 
sign  prefixed,  the  sign  plus  is  understood,  and  the  quantities 
are  said  to  be  positive  or  affirmative  ;  and  those  having  the 
sign  minus  prefixed  are  called  negative  quantities.  Thus, 
7.  =  -f-  0,  /)  =  +  6,  are  each  positive  quantities  ;  whilst 
—  a,  —  h,  are  negative  quantities.  When  the  sign  — ,  is 
prefixed  to  an  isolated  term,  as — a, — 6,  it  is  not  to  be 
considered  as  a  symbol  of  operation,  but  as  a  symbol  of 
condition,  merely  showing  that  a  and  &  are  in  a  state  or 
condition  directly  opposite  to  that  denoted  by  -|-  «  and  -{-  h. 
Thus,  if  the  degrees  of  the  thermometer  above  zero  are 
called  +,  then  those  below  must  be  called  — . 

(18.)  An  algebraic  expression  composed  of  two  or  more 
terms  connected  by  -f-  or  — ,  is  called  a  polyyiomial .  A 
polynomial  composed  of  but  two  terms,  is  called  a  hhio- 
mial ;  one  composed  of  three  terms,  is  called  a  trinomial. 


DEFINITIONS.  13 

Thus, 

3a  +  46,  ) 

7a:' — 3]/,  >  are  binomials. 
3o^ — x"',  ) 
3a=  +  46  —  x,  ) 
4771  —  y  +  a,  >  are  trinomials. 
5g    —  a;   +  3/,  5 
(19.)  Each  of  the  literal  factors  which  compose  any  terra, 
is  called  a  dimension  of  this  term  :  the  degree  of  a  term  is 
the  number  of  the  dimensions  or  factors.     Thus, 
7a,  )  are  terms  of  one  dimension,  or  of 
56,  ^  the  first  degree. 
5gx,  }  are  terms  of  two  dimensions,  or 
5a:]/,  ^  of  the  second  degree. 
la'P  =  laabbh,  ?  are  terms  of  five  dimensions,  or 
3x^     =  oxxxxx,  )  of  the  fifth  degree. 

(20.)  A  polynomial  is  said  to  be  homogeneous,  when  all 
its  terms  are  of  the  same  degree.     Thus, 

So  — 5a;  -|-  2y,  ^  are  homogeneous  polynomials  of 

b  —   y  -\-  m,j  \  the  first  degree. 
4ff*  +  2x^  —  jy,  ?  are  homogeneous  polynomials  of 
1am — c"  +  a^,  \  the  second  degree. 
5a"b^  —  6o^ — 4z'*y,    )  are  homogeneous  polynomials 
3a^6  —b^    -f-4a3/)2,  J  of  the  fifth  degree. 

(21.)  Any  combination  of  letters,by  the  aid  of  algebraic 
signs,  is  called  an  algebraic  expression.     Thus, 

_    ?  is  an  algebraic  expression,  denoting  seven  times 
'^  the  quantity  x. 

)  is  an  algebraic  expression,  denoting  that 
3  </  b  -\-  a"\  the  quantity  a  is  to  be  squared,  and  then 

)  added  to  three  times  the  square  root  of  h. 
(  4- na  ^ '^  an  algebraic  expression,  denoting  that 
^  '  '   ^  the  sum  of  a  and  b  is  to  be  squared. 

The  algebraic  expression  (a  -f  6)("  —  b)  =^  a"  —  /•',  is 
read,  the  sum  of  a  and  b,  multiplied  by  the  dilurence  of  a 
and  b.  equals  the  difference  of  the  squares  of  a  and  b. 


14  DEFINITIONS. 

(22.)  We  will  give  some  identical  algtbralc  expressions, 
which  may  serve  to  exercise  the  student  in  reading  algebraic 
formulas. 

(a-\-b)'^=a^-^2ab+b'.  (1) 

(a  — 6)*  =  a2  — 2a6  +  R  (2) 

{a  +  by  -^  {a  —  bY  =  2a'  +  2b-.  (3) 

i{a  +  b)-Ha-b)  =  b.  (5) 

Expression  (1)  is  read,  "  the  square  of  the  sum  of  a  and 
b  is  equal  to  the  square  of  o,  plus  twice  the  product  of  a 
and  6,  plus  the  square  of  6." 

Expression  (2)  is  read,  "  the  square  of  a  diminished  by  b 
is  equal  to  the  square  of  a,  minus  twice  the  product  of  a 
and  b,  plus  the  square  of  6." 

The  student  should  read  the  remaining  expressions  for 
himself,  and  should  also  form  other  expressions,  which  he 
may  in  like  manner  translate  into  common  language.  He 
should  also  substitute  particular  values  for  a  and  6,  in  the 
above  expressions,  and  see  if  the  results  on  both  sides  oi 
the  equations  are  identical. 

Thus,  the  above  expressions  become,  when 
a  :^  2,  and  6=1. 
(2  +  1^  =  4+4  +  1=9.  (1) 

(2_1)2==4  — 4  +  1  =  1.  (2) 

(2  +  1)3+  (2—1)  2=.  8  +  2  =  10.  (3) 

H2+l)  +  i(2-l)  =  2.  (4) 

U2+l)-H2-l)  =  l.  (5) 

If         a  =  J ,  and  6=1,  they  will  become 

(i  +  i)''=i  +  i  +  ^  =  H.  (1) 

U— ^r  =  i  — H-f=3V.  (2) 

0  +  ir  +  (5-i)^=i+f  =  lf.  (3) 

i(i  +  *)+KJ  — 0=1.  (4) 

Hh+^)-i{h-h)  =  h  _  _        (5) 

In  this  way  the  student  should  be  exercised,  until  he  be- 
comes familiar  with  the  nature  of  algebraic  expressions. 


ADDITION.  16 


ADDITION. 


(23.)  Addition,  in  Algebra,  is  finding  the  simplest  ex- 
pression for  several  algebraic  quantities,  connected  by  + 


Suppose  we  wish  to  find  the  sum  of 

We  first  seek  the  sum  of  the  positive  quantities,  by  pla- 
cing them  under  each  other  as  in  arithmetical  addition,  thus, 

+  3a'^6 


-f-  14a^6  =  sum  of  the  positive  terms. 

Proceeding  in  the  same  way  with  the  negative  terms,  we 
find 

—  lOa'b 

—  5a^b 

—  2a^h 


—  17a^6  =  sum  of  negative  terms. 

Therefore  the  total  sum  is  -|-  14a'6 — 17a'6=  —  3a"h. 

We  could  proceed  in  a  similar  way  for  expressions  of  a 
like  kind. 


16  ADDITION. 


CASE  I. 

(24.)  When  the  quantities  are  alike  but  have  unlike  signs, 
we  have  this 

RULE. 

I.  Place  the  different  terms  under  each  other,  add  the 
coefficients  of  the  positive  quantifies  into  one  sum,  and  the 
coefficients  of  the  negative  quantities  into  another. 

II.  Subtract  the  less  from  the  greater. 

III.  Prefix  the  sign  of  the  greater  sum  to  the  remainder, 
and. annex  the  common  letters. 

EXAMPLES. 

1.  What  is  the  sum  of 

2ahx  —  labx  —  2ahx  -\-  12ahx  -\-  abx  —  3abx  ? 
2abx 
12abx 
abx 


Idabx  =  sum  of  positire  terms. 


—  labx 

—  2abx 

—  3abx 


I2abx  =  sum  of  negative  terms. 


Therefore  15abx  —  12abx  =  Zabx  =  sum  total. 

2.  What  is  the  sum  of 

7amn  —  3ainn  -\-  2amn  +  5amn  —  lOamn  % 

Ans.  amn. 


ADDlTIOy.  17 

3.  What  is  the  sum  of 

■idaxy  —  31axy —  lOaxij  -\-  lOOaxy — laxij  -\-Aaxi/  ? 

Ans.  99axy. 

4.  What  is  the  sum  of 

3  Vox  -\-  7  ^/ax  —  5  y/ax  —  3  v/o.r+10  ^ax — 4  y/ax  ? 

Ans.  S-yax. 

5.  What  is  the  sum  of 

i  11 

ia^b  —  oab-  +41av/6  —  lab'  +  lOob'? 

Ans.  43a6^ 

6.  What  is  the  sum  of 

13ah'  +GJvh  —  20Jb^  +  lOah'  1 

Ans.  9ah\ 

7.  What  is  the  sum  of 

Ua^b-  Vx  +  aW'x'  —  20a-^b^x'  +  Ga'b"  \/x  ? 

Ans.  — 2a'^b~\/x. 


CASE   II. 

(25.)  When  both  quantities  and  signs  are  unlike,  or  some 
like  and  others  unlike. 

RULE. 

I.  Find,  the,  sum  of  the  like  terms  as  in  Case  I. 

II.  Then  write  the  su7ns  one  after  another,  loith  their 
•proper  signs. 

EXAMPLES. 

1.     What  is  the  sum  of 

Sax  —  2ah  +  4xy  —  2ax  +  3xy  -\-lab  —  2xy  -{-  6ax  7 
3 


18 


3ax 
-2ax 
Gax 

'7ux  =  sum  of  the  terms  containing 
—■2ah 


5ab  =  snm.  of  the  terms  containino;  ab. 


4:xy 
3xy 
2xy 

oxy  =  sum  of  the  terms  containing  xy. 

Therefore  lax  +  5ab  -\-  bxy  =  total  sum. 

2.   What  is  the  sum  of 

2d^x  —  Zax^  +  2a5  —  la^x  +  4.ax^  —  8a  6  —  6a=x 
-h  10aa:=  -}-  \2ab  ? 

2a" X —   3ax-  -|-    2fl6 

—  7a-x  +    dnx" —   Sah 

—  6a"a:+  lOax'^  +  12a6 


Ans.     — \\a~x-\-\\ax'^ -\-    &ab 

3.  4. 

SaSft*—   7a6*+5axy  4awi—   Sa;^"—   6a6 

■Id^b^ —   2aft'* —    axy  — lam-\-    4a7n,*  +      'i^ 

8a"5^  +      ab*  —  laxy  — 8am  —  lOam^ —    6a6 

a-h^  —  IQab*  +  3axy  am  +      am^  +  20a6 


5a268_18a6*  —10am—  8am2  +  9a6 


ly 


5.         •  6. 

1  s/y  —  4(a  +  m)  4a*  •+•    ban 

3v/y  —  2(a-j-??i)  — 3«- —    Ian 

s/y  +  7(rt  -}-  '^0  2fl- —   3a7i 

5  s'y  +    ('''  +  //' )  5«-  +  lOovi 


16 ^///  +  2(a  +  m)  8a^  4-    5a/i 


7.  8. 

a{a-^h)-\-  3v^fl— X  3xy  +  2m^/a+    3 


~4'.(«-f?,)— lOv^a- 


.r  y 

-|-  Im  \/  « — 

7 

I3x"^y 

—   m-^/a  — 

10 

-2.^;v. 

+  3/71  \/a  + 

2 

ox   ?/- 

—  4w\/a  — 

1 

21x*y 

—  7w  v/a  — 

13 

-7fl(a-t-6) —  4v'a  —  x 


9. 

i    i   ,         1    ^  10. 

x^m^+      xjrn^  26+    8x  +  6y 

-Wx'.in-—  6x"'m'  36—   Sx  — 4y 

14x^;;i^+    5x^m^  —76—    3x-    y 

1    ^            J-    -i  86  +  10x  +  3.y 

— 7x"*m"  —  12x"77i''  oj, 

9o  —      X  —    y 

_33:V^_12x^m'  15/>  +    6x  +  3y 


1 1 .  What  is  the  sura  of  "^ag  -j-  6am  —  9xy  +  3a6  —  xy 
-f  4«o-  +  10am  —  7xy  —  6a6  +  5xy  +  4a o-  —  l3om  ? 

Ans.  Wag  -\-  Sam  —  12xy  —  3a6. 


20  ADDITION. 

12.  What  is  the  sum  of  'ia'^x  —  5a^y  -{-lam — Sa"x 
—  lOa^y  —  .iam  +  da^y  —  lirx  —  13am  +  Ca'y  —  Wa'x 
-f-  am  —  a^y  -\-  a'^^y  —  Ga'^x  1 

Ans.  — 23n^x  —  9am. 

13.  What  isthesumof — 2>xy  -\-bn  -\-  3ax —  \Oam  —  6xy 
-f  In  —  -iax  -\-  Sam  —  2xy  +  lO?^  +  Gam  —  4ax  ? 

Ans.  — 11 X y -\- 22n  —  5fla: -f- 4cw. 

14.  What  is  the  sum  of  4a-  +  5a-6V^  —  9a^  +  6a^6'-c- 
4-  lOa'x  +  Ta^o:  +  Scr—  I3a^6^c^  +  oa"  —  3a^x  +  Sa^bh'  1 

Ans.  8a*  +  a-&V  +  14a3x. 

15.  What  is  the  sum  of  ^a  —  'ixy — V^  —  n -\- %xy 
+  5  v^a  4-  7n  —  7xy  +  9  v'o  —  7  Vni  +  16;?  —  5  v/a  ? 

Ans.  lOv/fl— 4x7/  — 8  Vin  +22n. 

16.  What  is  the  sum  of  6a   v^6  -\-bx^y   — 7x  +  5a  h 
-h  6x  — 3a*6^  — 4x2  y^  _|_  2a\/6  — lOx  +  8a*6*  ? 

Ans.  l8«*6'^H-x^y*— llx. 

17.  What  is  the  sum  of  3  \^a-\-h — 5  ^/x-\-bxry^-\-l  ^ a-{-h 
4-3x^^3  — 7  s/x-f  y/  fl-f  6  —  8x'2/"  +  2  y/x  4-  10  VT+l  ? 

Ans.  21v/a4-6  — lOv/x. 

18.  What  is  the  sum  of  ba^h-c  —  4a6ac^  4-  2a26-c*  —  7ax 
4-  Qa^Pc  —  baVc^  —  13ax  —  70^6=02  4-  3a6'^c'  —  Sa^J'c  ? 

Ans.  3a'62j,  „  ^^52^3  _5a-.'62(.2  _  gOox. 


SIBTRACTIOX.  21 


SUBTRACTION. 

(26.)  SuBTRACTiox,  ill  Algebra,  is  the  finding  the  sim- 
plest expression  for  the  ditference  of  two  algebraic  expres- 
sions. 

If  we  subtract  the  positive  quantity  b  from  a,  we  obvi- 
ously obtain 

a  —  b, 
which  is  the  same  as  the  addition  of  a  and  — b. 

Again,  if  we  wish  to  subtract  b  —  c  from  a,  we  obtain 
by  subtracting  6  from  a,  a  —  6,  but  we  have  subtracted  too 
much  by  the  quantity  c,  therefore  adding  c,  we  get 

a  —  6  +  c, 
which  is  the  same  as  the  addition  of  a  and  —  b  -{-  c. 

From  this,  we  see  that  subtracting  a  quantity  is  the  same 
as  adding  it  after  the  signs  are  changed. 

Hence,  for  the  subtraction  of  algebraic  quantities  we 
have  this 

RULE. 

I.  Write  the  terms  to  be  subtracted  wider  the  similar 
terms  y  if  there  are  any^  oy  those  from  which  they  arc  to  be 
subtracted. 

II.  Conceive  the  signs  of  the  terms  of  the  polynomial  to 
be  subtracted,  to  be  changed,  and  then  proceed  as  iji  additian. 


29  SUBTRACTION. 


EXAMPLES. 


1.  2. 

From  lac —   Sab -{-    d-       From  Samx —    4-^2/+    5y^ 
Take   4ac  +    Sab  +  id'       Take    damx+lOxy—Uf 


Rem.  3ac  —  Uub  —  3d-  Ri.'m.  —  amx—Uxy-]-l6y^ 

3.  4. 

From  6a:y  —  3ac  +    2,71^  From4o^v/x- — oa  i/y -\-    x 

Take  4xy  —  lac  —    9m"  Take  3<r''  ^/x  +  3a  Vy  —  ""'^ 


Rem.  2a:y  +  4ac  +  Ib/i^       Rem.     a^ v^o: — 8a  Vy  +  8^ 


5.  From       3a"6c  —  7oxy-|- 37ny -f- "  take  d'bc  -\-'$>axy 
+6a   ~4:my. 

Alls.  2a-6c  —  I5axy  -\-  Imy  —  5a. 

6.  From  9>abVc — I2a^h -\-(icx  —  Ixy  take  9a6v/c 

—  I3a^6  +  Sxy  —  an  -\-  2cx. 

Ans.  —  ab^c  -\-  a^b  +  2cx  —  Ibxy  +  an. 

7.  From  Iba^x  —  14a-y  +  3a6'  +  Qamn  take  Qmg  +  3a 

—  ba^x  —  7a-y  -\-?>ab^  —  4flwm  —  4. 

Ans.  20a^x  —  7a^y  +  lOam^t  —  Q)iiig  —  3a  +  4. 

8.  From  I3a^a;^y  +  3a.r  —  lab  -\-  Qmg  —  x'^y'^  take  bxy 
+  4aV2/  —  Qax  +  9aA  +  2mg  +  bxhf. 

Ans.  ^a*x^y  +  9c3-  —  16a6  +  4mo-  —  Gary^  —  bxy. 

1    1 

9.  From  7a-6-'-|-3a~  —  4i''x  —  3ax//-f-4x- — 3x?/^take 

3a6  — 17  +4a2_5a'6i  —  7i^j-  +  3xy^ 

Ans.  12a^6^—a-'4-3//'a—3afi/4-4a;-— 6x2/3— 3a6  +  17. 

10.  From     4a^ix  —  7axy  +  3//'7  +17  —  x    take   4,p'^q 

—  13  -|-  7a^6x  +  Suxy  —  7x  +  3/ —  mg  -\-  n. 

Aus.  — 3a^i'X —  15ajy  —  jrq  +30  +6x  —  3/+  mg  —  n. 


SIBTRACTIOX.  23 

1 1 .  From  6am  -\-  x  take  Sam  -|-  y. 

Ans.  3am  -\-x  —  y. 

12.  From  3a'm — Qx-y'^^2xy  take  4:(rm-\-  6x^7/^ -j-o^y. 

Ans.  — a"m  —  I2xh/  — 2xy. 

13.  From  "^amx  —  43  -j-  x — y-f-27d  take  15/i  +  Ig  —  3 
4-4]/  —  Sd-\-  lamx  —  x-\-p(l  —  rs. 

Ans. — ^amx — 40-}-2a: — by-\-2>bd — Ion — Ig — jjq-j-^s. 

14.  From  a  -{-  b  take  a  —  b. 

Ans.  2b. 

(27.)  We  ean  express  the  subtraction  of  one  polynomial 
from  another,  by  writing  the  polynomial  which  is  to  be  sub- 
tracted, after  enclosing  it  within  a  parenthesis,  immediately 
after  the  other  polynomial  from  which  it  is  to  be  subtracted, 
observing  to  place  the  negative  sign  before  the  parenthesis. 

Thus,  ab  —  Gxy-j-  3am  —  (4«/;  -f-  3xy  -)-  am) 
denotes,  that  the  polynomial  enclosed  within  the  parenthesis 
is  to  be  subtracted  from  the  one  which  precedes  it ;  and  since, 
by  (Art.  26),  to  perform  subtraction,  we  must  change  all  the 
signs  of  the  terms  to  be  subtracted,  we  may  remove  the  pa- 
renthesis provided  we  change  the  signs  of  the  terms  which  it 
encloses :  and  conversely,  we  may  enclose  any  number  of 
terms  within  the  parenthesis,  with  a  negative  sign  before  it,  if 
we  observe  to  change  the  signs  of  the  terms  thus  enclosed. 

In  this  way  we  can  transform  the  expression 
a'b-\-xy  —  1am — {7nx-\-6 —  iSx*), 


mto 
into 
into 
into 


a'b-\-xy  —  lam —  7nx  —  6-j-  I3x-, 
a%-\-xy  —  {lam-j-mx-^G —  13a;-), 
a^6  —  ( — xy-\-1am-\-mx-\-6  —  iSx"), 
a'^b-{-xy  —  lam  —  mx  —  (6  —  I3x'). 


94  MULTIPLICATION. 


MULTIPLICATION. 

(28.)  If  we  wish  to  multiply  a  by  6,  we  must  repeat  a  as 
many  times  as  there  are  units  in  &,  which,  by  (Art.  6),  is 
done  by  writini^  h  immediately  after  a,  thus,  a  multiplied 
by  ft  =  ab. 

Again,  if  we  wish  to  multiply  a  by  —  5,  we  obserA'e  that 
this  is  the  same  as  to  multiply  —  h  by  a,  hence  we  must  re- 
peat —  6  as  many  times  as  there  are  units  in  a:  repeating  a 
minus  quantity  once,  twice,  thrice,  .or  any  number  of  times 
can  not  change  it  to  a  positive  quantity.  Hence,  —  h  multi- 
plied by  c,  or,  which  is  the  same,  a  multiplied  by — 6= — ah. 

Finally,  if  we  wish  to  multiply  a  —  6  by  c  —  (/,  we  will 
first  multiply  a  —  6  by  c,  we  thus  obtain 
a  —  6 


ac  —  6c  for  a  —  h  repeated  c  times. 
This  result  is  evidently  too  great  by  the  product  of  a  —  h 
by  d,  since  it  was  required  to  repeat  a  —  6  as  many  times  as 
there  are  units  in  c  lessd. 

Then  repeating  a  —  6  as  many  times  as  there  are  units  in 
rf,  we  have 

a —  6 
d 


ad  —  hd  for  a  —  b  repeated  d  times. 
Subtracting  this  last  result  from  the  former,  we   have 
ac  —  be  —  {ad  —  bd),  which,  by  (Art.  27),  becomes 
ac  —  be  —  ad  -j-  bd  for  the  product  of  a  —  b  by  c  —  d. 


MTLTIPLICATIOX.  25 

Hence,  we  see  thai  —  b,  when  multipiied  by  —  d,  product  s 
the  product  +  bd 

If  we  wished  to  multiply  a  by  —  6,  it  would  hardly  be 
correct  to  say,  that  we  are  to  repeat  a  mi7ius  b  times  ;  for 
a  quantity  cannot  be  repeated  a  7ninus  number  of  times. 
But  when  we  wish  to  multiply  a  by  —  6,  we  evidently 
wish  to  repeat  a  as  many  times  as  there  are  units  in  6,  and 
then  to  give  to  the  product  the  negative  sign  ;  that  is,  when 
the  multiplier  is  negative,  we  must  multiply  as  though  it 
were  positive,  and  then  give  to  the  product  a  contrary  sign. 

Applying  this  principle  to  the  case  of  —  a  multiplied  by 

—  b.  We  know  that  —  a  multiplied  by  -|"  ^  gives  —  ab 
for  the  product ;  therefore  —  a  multiplied  by  —  b  must 
give  the  same  product  taken  with  a  contrary  sign  ;  that  is, 

—  a  multiplied  by  —  b  must  give  +  o^- 

(29.)  From  all  this,  we  discover,  that  the  inoduct  will 
have  the  sign  +,  iL'hen  both  factors  have  like  signs,  and  the 
product  will  have  the  sign  — ,  lohen  the  factors  have  con- 
trary signs. 

If  we  wish  to  multiply  3a-6  by  4a'6-,  we  observe  that 

3a'b  =  Saab 

4a^b~  =  4:aaabb 

Hence,  the  product  will  be 

2aab  X  -iaaabb  =  12aaaaabbb  =  12a^b^. 

Here  we  discover  that  the  exponent  of  a,  in  the  product, 
is  equal  to  the  sum  of  the  exponents  of  a  in  the  factors  ; 
likewise  the  exponent  of  fe,  in  the  product,  is  equal  to  the 
sum  of  the  exponents  of  b  in  the  factors. 

(30.)  Hence  the  product  of  several  letters  of  different  ex- 
ponents is  equal  to  the  product  of  all  the  letters,  having  for 
exponents  the  sums  of  their  respective  exponents  inJhe  fac- 
tors. 


26  MULTIPLICATION. 

CASE  I. 

(31.)  From  what  has  been  said,  we  have,  for  multiplying 
this 

RULE. 

I.  Multiply  the  coefficients^  observing  to  prefix  the  sion  -J- 
whcn  both  factors  have  like  signs;  and  the  sign  -r-  when  they 
have  contrary  signs. 

II.  Write  the  letters  one  after  another;  if  the  same  letter 
occur  in  both  factors^  add  the  exponents  for  anew  exponent. 

(32.)  The  product  will  be  the  same  in  whatever  order  the 
letters  are  placed,  but  it  will  be  found  more  convenient,  in 
practice,  to  have  a  uniform  order  for  their  arrangement. 
The  order  usually  adopted  is  to  place  them  alphabetically. 

EXAMPLES. 

1.  Multiply  llax'^y  by  3ay. 

Ans.  33aV2/''. 

2.  What  is  the  product  of  3a7n^  by  Qa'^b^x  ? 

Ans.  iSd^Wiivx. 

3.  What  is  the  product  of  \OcH^  by  ^ahdl 

Ans.  'dQah'd^ 

4.  Multiply —  13ac^  by —40^6 V. 

Ans.  b2a^b'^c^. 


5.  Multiply  a'Vc''  by  a''b\ 

6.  Multiply  —  11  xhj  by  ?>xyz. 


Ans.  a'"^"h"^'c''. 


Ajis.  — blx^y-z. 


Mamply  -ab'^cd  by  -axy.  3 

4  '  Ans      — n- 


Ans.   -—a-h^cdxy. 
28  ^ 


MULTIPLICATION.  27 

8.  Multiply  —-xyzhy~xYz\  j 


2  "      '2 
9.  Multiply  Im^n^p"^  by  Gmn'^p^ 


Ans. xV-^ 

4    -^ 


CASE   II. 

(33.)  Polynomials  may  be  multiplied  together  by  the  fol- 
lowing 

RULE. 

I.  Multiply  all  the  terms  of  the  multiplicand  succes- 
sively by  each  term  of  the  multiplier,  and  observe  the  same 
rules  for  the  signs  and  exponents  as  in  Case  I. 

II.  When  there  arise  several  partial  products  alike,  they 
must  he  placed  under  each  other,  and  then  added  together 
in  the  total  product. 

(34.)  The  total  product  will  be  the  same  in  whatever 
order  we  multiply  by  "the  terms  of  the  multiplier,  but  for 
the  sake  of  order  and  uniformity,  we  begin  with  the  left- 
hand  term. 

EXAMPLES. 

1 .  What  is  the  product  of  Sa'  —  6a.T  -f-  y  by  3a  —  m  ? 

OPERATION. 

3a2  —  dax  -|-  y 
3a  —  m 

Ans.  9a'  — iSa^  x-\-Zay  —  3a2  m  -{-  Qamx  —  my . 


i  MULTIPLICATION. 

2.  What  is  the  product  of  Cx^— Sy'-f-a  by  a;«— 2y»— a  ? 

OPERATION. 

6x2—    32/3     +  a 
X-  —   2y^     —  a 


6x*—   3xy+  0x2 

—  12x-y^ — 6ax--\-6y^ — 2ay ' 

-{-2ay^ — a^ 


Ans.  6x^— 15xy  — 5ax-+6/-f  ay^—a\ 

3.  What  is  the  product  of  b~m  — Say  by  6x  —  3  ? 

Ans.  66~mx —  ISaxy —  db"m  -\-  9ay. 

4.  What  is  the  product  of  7/— 2m— 9  by  3/— 11m  ? 

Ans.  21/'— 83/m— 27/4-22;?i-+99m. 

5.  Multiply  2a-\-ol+3i—5e  by  3a +  106 +  15/ 

.        {  6a=2+35a6+9ac— 15ae+50/>-+306r 
^"s.  I  _50ic+30rr/+75//+45c/— 75e/. 

6.  Multiply  a  +  6  +  c+d  by  a  —  J- c  —  (/. 

Ans.  a~  —  b'  —  2bc  —  26d  —  c'  —  2c(Z  —  d^ 

7.  Multiply  a^-\-a*-{-  a-  by  a-  —  1. 

Ans.  a  —  (I-. 

8.  Multiply  a^  -f-  flz  +  z-  by  a-—az-{-  z". 

Ans.  o^  +  a"z^  +  c''. 

9.  Multiply  a  +  6  by  a  +  6. 

Ans.  a=  +  2fl6+6^ 

10.  Multiply  a  —  5  by  a  —  b. 

Ans.  a'  —  2ah-\-b'\ 

11.  Multiply  a  +  6  by  a  —  b. 

Ans.  a*  —  6*. 

(35.)  The  last  three  examples,  when  translated  into  com- 
mon language,  give  three  distinct  and  important  theorems, 
which  we  will  proceed  to  illustrate. 

Example  9  is  the  same  as 

(a  +  5)  X  (a  +  ft)  =  (a  +  bf  =  a^  +  2ab  +  b-  ; 
which,  when  translated,  gives 


MULTIPLICATION,  29 

THEOREM   T. 

The  square  of  the  sum  of  two  quantities  is  the  same  as  the 
square  of  the  first ^  plus  twice  the  product  of  bothj  plus  the 
square  of  the  secofid. 

EXAMPLES. 

1.  (x+y)  X  {x-\-y)={x-j-yy=x  Jr2xy-\-y\ 

2.  (2x-f-fl)  X  (2x-j-a)=(2.r-)-a)-=4z-+4ax-(-a*. 

3.  (5m+3)  X  (5/7i-}-3)=:  (5/n4-3)==25/?i-+3077i+9. 

Example  10  is  the  same  as 

{a—b)  X  {a—b)=^{a—by-:=^a'--2ab+b^ ; 
which,  when  translated,  gives 

THEOREM    II. 

The  square  of  the  dijference  of  two  quantities  is  equal  to 
the  square  of  the  first,  minus  twice  the  product  of  both, plus 
the  square  of  the  second. 

EXAMPLES. 

1.  {x  —  y)x{x  —  y)={x  —  y)-^x^—2xy  +  y\ 

2.  (3a  —  6)  X  (3a  —6  )  =r  (3a  —  &)=  =9a'  —  Gab  +  b^. 

3.  (5a  — x)X(5a— a;)  =  (5a  — x)'  =  25a'  — lOax+i'^ 
Example  11  is  the  same  as 

{a  +  b)X{a~b)  =  a'—h-', 
which,  when  translated,  gives 

THEOREM    III. 

The  sum  of  two  quantities  multiplied  by  their  difference, 
is  equal  to  the  square  of  the  greater,  minus  the  square  of 
the  less. 

EXAMPLES. 

1.  {x  +  y)x{x-y)=ar'  —  y\ 

2.  (3a  +  6)x(3a  — 6)  =  9a2— 6-. 

3.  (7m  -f-  y)  X  (Jm  —y)=  49wr  —  f. 


30 


DIVISION. 

(36.)  We  know  by  the  principles  of  Arithmetic,  that,  if, 
in  Division,  we  multiply  the  divisor  into  the  quotient,  tli'' 
product  will  be  the  dividend. 

Therefore,  referring  to  what  has  been  said  under  Multi- 
plication (Art.  29),  we  infer  that  when  the  dividend  has  the 
sign  -f-j  the  divisor  and  quotient  must  have  the  same  sign  ; 
but  when  the  dividend  has  the  sign  — ,  then  the  divisor  and 
quotient  must  have  contrary  signs. 

(37.)  Hence,  when  the  dividend  and  divisor  have  like  sigiia, 
the  quotient  will  have  the  sign  +;  and  .when  the  dividend 
and  divisor  have  contrary  signs,  the  quotient  loill  have  the 
the  sign  — . 

We  have  also  seen  under  Multiplication  (Art.  30),  that 
the  product  of  several  letters  of  different  exponents  is  equal 
to  the  product  of  all  the  letters  with  the  sum  of  their  re- 
spective exponents  for  new  exponents. 

(38.)  Hence,  to  divide  any  power  of  a  letter  by  a  different 
power  of  the  same  letter,  it  is  obvious  that  the  quotieid  will 
be  a  power  of  the  same  letter,  having  for  exponent  the  ex- 
cess of  the  exponent  in  the  dividend  above  that  of  the  divisor. 
(39.)  If  we  divide  continually  the  expression 
a^  =  aaaaa  by  a,  we  shall  find 
a^  -7-  a=:a^~^  =  a*  =  aaaa ; 
a'^  -r-  a=a'^  —  ^  =  a^  =  aaa  ; 
a*  -T-  a  =  a^  ~  '  =  a'^  :=  aa  ; 
a^  -r-  a  =  a^~^  =  a^  =  a; 
a*  -7-  a  =  fl' " '  =  a"  =  1  ; 


i 


DIVISION.  31 

'=-=reciprocal  of  a; 
a 

a~'-ra=a"~'~'=a~-=    -=— ^reciprocal  of  a*; 
a  a      a~ 

a—'-r-a=a~'~^=a~^-^ =— =reciprocal  of  a^ ; 

aaa     a" 

a-^-x-a=a~^~^^a~^= =— =reciprocal  of  a*  ; 

aaaa     a* 

&c.  &c. 

(40.)  From  the  above  scheme,  we  see,  that  whenever  the 
exponent  of  a  quantity  becomes  0,  its  value  is  reduced  to  1 . 

(41.)  That  whenever  it  is  negative,  it  is  the  reciprocal  of 
what  it  would  he  were  it  poaitive. 

(42.)  Hence,  changing  the  sign  of  the  exponent  of  a 
quantity  is  the  same  as  taking  its  reciprocal. 

CASE   I. 

(43.)  From  what  has  been  said,  we  have,  for  dividing  one 
monomial  by  another,  this 

RULE. 

L  Divide  the  coefficient  of  the  dividend  by  that  of  the 
divisor,  observing  to  prefix  to  the  quotient  the  sign  -\-  when 
the  signs  of  the  dividend  arid  divisor  are  alike,  and  fhc 
iign  —  when  they  are  contrary. 

II.  Subtract  the  exponents  of  the  letters  in  the  divisor 
from  the  exponents  of  the  corresponding  letters  of  the  divi- 
dend J  if  letters  occur  in  the  divisor  ichich  do  not  in  the 
dividend,  they  may  (Art.  42)  be  written  in  the  quotient  by 
changing  the  signs  of  their  exponents. 

(44.)  It  must  be  recollected  here,  and  in  all  cases  here- 
after, that  when  the  exponent  of  a  letter  is  not  written,  1  is 


32  DIVISION- 

always  understood  (Art.  12)j  and  when  the  exponent  is  0, 
the  value  of  the  power  is  1.  (Art.  40.) 

EXAMPLES. 

1.  What  is  the  quotient  of  14a^a;°  divided  by  lax^y  ? 
Dividing  the  coefficients  we  find  2,  to  which  if  we  annex 

the  letters  after  subtracting  the  exponents,  we  have 

the  X  has  disappeared,  since  its  exponent  became  0,  and  its 
value  therefore  was  1,  by  (Art.  40.)  And  since  the  y  oc- 
curred in  the  divisor  and  not  in  the  dividend,  it  was  written 
in  the  quotient  with  the  sign  of  the  exponent  changed. 
(Art.  42.) 

2.  What  is  the  quotient  of  Sof.-'^Pc  divided  by  bahcl 

Ans.  lah"^. 

3.  What  is  the  quotient  of  — 44;n7ix*  divided  by  22  ahcx  ? 

Ans.  — 2a-^'b~^c~^mnx. 

4.  Divide  — Ix'^y  by  lOx^y. 


5.  Divide  Za^rri^n^  by  — Qamn. 

6.  Divide  35a;^2/cMjy  —  Txyz^ 

7.  Divide  cd}'  by  — 13cd*^ 


8.  Divide  — 3a"'6"  by — 4a''6  c\ 

9.  Divide  —llaH^'m-^  by  -^a^H^ 


Ans. 

10 

Ans.  — 

1   .    ■»  ', 

Ans. 

—bxy-'\ 

Ans. 

-Id-K 
13 

A         3  „ 

Ans.  -a" 
4 

'~'ir-^c-\ 

'm-\ 

Ans 

4 

DIVISION.  33 

10.  Divlile  13j-'?/-^  by  —  26x2/. 

Ans. a—' J/-®. 

2      '^ 

(45.)  To  divide  one  polynomial  by  another,  we  shall  imi- 
tate the  arithmetical  method  of  long  division.  And  in  the 
arrangement  of  the  work  we  shall  follow  the  French  method 
of  placing  the  divisor  at  the  right  of  the  dividend.  Thus, 
to  divide 

a^  4-  a'^x  -\-  ub  -\-  bx  by  a-\-x, 

we  proceed  as  follows  : 

OPERATION. 

Dividend  =  a"  -(-  a'x  -\-  ab  -\~  bxla   -j-  x  =  divisor 

a^  -f-  a'^x  I 

a^  -\-  b  =  quotient 


ah  +  bx 
ab  -\-hx 

0 
Having  placed  the  divisor  at  the  right  of  the  dividend,  we 
seek  how  many  times  its  left-hand  term  is  contained  in  the 
left-hand  term  of  the  dividend,  which  we  find  to  be  a'-*,  which 
we  place  directly  under  the  divisor,  and  then  multiply  the 
divisor  by  it,  and  subtract  the  product  from  the  dividend  ; 
then  bringing  down  the  remaining  terms,  we  again  seek  how 
many  times  the  left-hand  term  of  the  divisor  is  contained 
in  the  left-hand  term  of  this  remainder,  which  we  find  to 
be  b  ;  we  then  multiply  the  divisor  by  b,  and  again  subtract- 
ing there  remains  nothing  ;  so  that  n'^-\-h  is  the  complete 
quotient. 

That  the  operation  may  be  the  most  simple,  it  will  be 
necessary  to  arrange  both  dividend  and  divisor  according  to 
the  powers  of  some  particular  letter,  commencing  with  the 
highest  power. 


34 


CASE   II. 

(46.)  To  divide  one  polynomial  by  another,  we  have  this 

RULE. 

I.  Arrange  the  dividend  and  divisor  with  reference  to  a 
certain  letter;  then  divide  the  first  term  on  the  left  of  the 
dividend  by  the  first  term  on  the  left  of  the  divisor,  the  re- 
sult is  the  first  term  of  the  quotient;  multiply  the  divisor 
by  this  term,  and  subtract  the  product  from  the  dividend. 

II.  Then  divide  the  first  term  of  the  remainder  by  the  first 
term  of  the  divisor,  which  gives  the  secon  '  term  of  the  quo- 
tient ;  multiply  the  divisor  by  this  second  term,  and  subtract 
the  product  from  the  result  after  the  first  operation.  Con- 
tinue this  process  until  we  obtain  0  for  remainder;  or  lohen 
the  division  does  not  terminate,which  is  frequently  the  case^ 
we  can  carry  on  the  above  process  as  far  as  we  choose,  and 
then  place  the  last  remainder  over  the  divisor ,  forming  a 
fraction,  which  must  be  added  to  the  quotient. 

EXAMPLES. 

1.  What  is  the  quotient  of  2a'b  -\- b^ -\- 2ab^ -\- a'  di- 
vided by  a"  -\- b^  ■]-  ab  1 

Arranging  the  terms  according  to  the  powers  of  a,  and 
operating  agreeably  to  the  above  rule,  we  have 


Dividend  =  a'  +  2a''b  +  2a6^  +  b 


OPERATION. 

a^  -|-  ab  -{-b^  =  divisor 


a  -\-b  =  quotient 


„2^+    ab^  +  b^ 
a''b+    a6'  +  6' 

0 


DIVISION.  35 

2.  What  is  the  quotient  of  a=i— Sa^-f-Safe— 6«— 46-f22 
iliviiledby  6  —  3? 

OPERATION. 


10 

6—3 


a-b- 
a-b- 

-3a' 
-Zir 

+'2ab- 

_6g— .li-l-92  b 

a 
-6a 
-6a 

—3 

^_|_2fl— 4  + 

2ab- 
2ab- 

—46+22 
— i6-fl2 

10= 

^remainder. 

3.  Divide  x^—x*-{-x'—x''-\-2x—l  by  x^  -f-x— 1. 


OPERATION. 


:j:^^x*+x'—x'i-2x—l 
x^-\-x'—x* 

— x^-{-x^ — x'^ 

x*—x'+2x 
x'+x'—x^ 

— x'+2x— 1 
— x^ — x^4-x 

x"-\-x — 1 
x^-f-x— 1 


X-  +  X — 1 


:4_a;3_^a.2_3._|.i 


4.  What  is  the  quotient  of  x' — Sax^-j-Sa'x — a^  divided 
? 


36 


OPERATION. 


X'  —  3ox-  +  3a'x  —  o3'x  ■ 

X3 QTT 


\x^  —  2ax  -\-  a'^ 


—  2ax"  -f-  3a'x 

—  2ax^  4-  2a''x 


a^x  —  a" 
a^x  —  a^ 


0 


5.  Divide  Uaf—  21hf-\-  lcf+  Sag  —  %g  -j-  3cg  by 
7/+ 3  a-.  Ans.-2a  — 36  +  c. 

6.  Divide  4x3  +  4x2  — 29x  +  21  ^^  2x  — 3. 

Ans.  2x--|-5x  —  7. 

7.  Divide  119c2  — 200cf/-|-408ce— 113cA  — 39c/=^ 
-f-  72de  +  31  dh  —  96e/i  +  20h'  by  17c  +  3d  —  4/i. 

Ans.  7r  — 13(?  +  24c  — 5/i. 

8.  Divide  72x*  —ISx'y  —  lOx^.v*  +  Hxy^  -|-  Sy^  by 
Gx'^  —  4x1/  —  J/?. 

Ans.  12x2  —  5xy  —  3y2. 

9.  Divide  36aH  —  e3ab^-}-20b^  by  12o6  — 5/r. 

Ans.  3a— 46. 

10.  Divide  a^  —  b'-  by  a  —  b. 

Ans,  a  -{-b. 

11.  Divide  a^  —  6^  by  a  — 6. 

Ans.  a^  +  a"b-i-ab"-\-bK 

(47.)  The  following  examples  cannot  be  accurately  per- 
formed, there  being  still  a  remainder,  however  far  the  divi- 
sion be  carried. 


12.  Dividing  1  by  1  —  6,  we  have  in  succession 
b 


1^(1-6)  =  1  + 


1  —  6 

b' 


='^'+l-b 
=  1  +  6  +  6-^+     ^ 


=  1  +  6  +  6=  +6 


1  — & 
b' 


1  —  6 

&c.  &c. 


,3.       1->(1  +  6)  =  1--A-^ 


l_6-{_52_63 


1  +  6 

6^ 


1+6 
&c.  &c. 


U.    (l+a:)^(l-a:)  =  l+    ^"^ 


=  l+2x-i 


l—x 

2x^ 


l  —  x 

2x^ 


l+2x  +  2x"  + 


l  —  x 


37 


=  1  +  2x  +  2x'  +  2x3+    ^^' 


1— :r 
&c.  &c. 


38  ALGEBRAIC    FRACTIONS. 


CHAPTER  tt 


ALGEBRAIC  FRACTIONS. 


(48.)  In  our  operations  upon  algebraic  fractions,  we  shall 
follow  the  corresponding  operations  upon  numerical  frac- 
tions, so  far  as  the  nature  of  the  subject  will  admit. 

CASE  I. 

To  reduce  a  monomial  fraction  to  its  lowest  term,  we 
have  this 

RULE. 

I.  Find  the  greatest  common  measure  of  the  coefficients  of 
the  numerator  and  denominator.     (See  Arithmetic.) 

II.  7%en,  to  this  greatest  common  measure  annex  the  let- 
ters which  are  common  to  both  numerator  and  denominator^ 
give  to  these  letters  the  lowest  exponent  which  they  have, 
whether  in  the  numerator  or  denominator  ;  the  result  will 
be  the  greatest  common  measure  of  both  numerator  and  de- 
nominator. 

III.  Divide  .  both  numerator  find  denominator  by  this 
greatest  common  measure^  (by  Rule  under  Art.  43,)  and  the 
resulting  fraction  will  be  in  its  lowest  terms. 


ALGEBRAIC    FRACTIONS.  39 


^     „    ,        315a^bxy    ,    .,    , 

1.  Reduce '--  to  its  lowest  terms. 

The  greatest  common  measure  of  373  and  15  is  15,  to 
which  annexing   abxy,   we  have   loabxy   for  the  greatest 
common   measure    of   both    numerator    and    denominator. 
Dividing  the  numerator  by  Ibabxrj^  we  find 
yiba^bxy  -i-  loabxy  =  25a^. 
In  the  same  way  we  find 

loab'^xy^  -r-  loabxy  =  by- ; 
hence,  we  have 

2>1ba^bxy^ 25a- 

loab-xy^         by' 
which,  by  Rule  under  Art.  44,  becomes 

— -  -  =  2oa-b-^y-'. 
by' 

-    ,,    ,        42ax'^7/r''      .     , 
2.  Reduce  „-;: — ;^to  its  lowest  terms. 
3oxy^z^ 

In  this  example,  the   greatest  common   measure  of  the 

numerator  and  denominator  is  Ixyz^  ;  hence,  dividing  both 

numerator  and  denominator  of  our  fraction  by  Ixyz^,  we 

find 

4:2ax'yz'       Gax'z''      ,  •  ^  •    •     .,    , 

— — r^  ==::  —,-~r-i  which  IS  in  its  lowest  terms. 

35xy^z^  oy' 

_    „    ,         — iSmnx^y^  ^    .     , 

6.  Reduce :-  to  its  lowest  terms. 

12/nxY' 

Ans.  —  — . 


1 3x^ 
4.  What  is  the  simplest  form  of  — — —  1 
^  26xy^ 


2y* 


40  ALGEBRAIC    FRACTIONS. 

5.   What  IS  the  simplest  form  of ? 

Ans.  9/>-fi«. 

(49.)  Fro??i  iij/ia^  has  been  said  (Art.  42),  we  infer  that 
we  may  transfer  a  letter  from  the  numerator  to  the  denomi- 
nator, or  from  the  denominator  to  the  numerator,  by  changing 
the  sign  of  the  exponent. 

Thus, 

,       laxyz         7z       ^       o     .            7 
1.      f-r-=  — ^  =7ca:-"v~   = : — r- 

„      1^     ,  ,     ,       17x^         17 


49aic^       ..      .   • 
■35^ 
all  the  letters  to  the  numerator. 


3.  Reduce  ^^  ^      to  its  simplest  terms   and  then  transfer 


7c«       7 


4.  Reduce  in  a  similar  manner  the  fraction 


lOSaSft^ 


Ans.  — ^^ —  =  -a~'b~^cdm~'^ . 
4ia-bhn      4 


GREATEST  COMMON  MEASURE  OF  POLYNOMIALS. 

(50.)  Before  proceeding  to  the  reduction  of  polynomial 
fractions,  it  is  necessary  to  show  how  to  find  the  greatest 
common  measure  of  two  polynomials,  which  may  be  effected 
by  this 

RULE. 

Divide  one  of  the  polynomials  by  the  other,  and  the  pre- 
ceding divisor  by  the  last  remainder,  till  nothing  remains ; 
the  last  divisor  will  be  the  greatest  common  measure. 

This  rule  may  be  demonstrated  as  follows  : 


ALGEBRAIC    FRACTIONS.  41 

(51.)  Let  JV  and  /i  be  two  polynomials,  of  Avhich  .Yis 
greater  than  n ;  then,  performing  the  divisions  as  directed 
in  the  above  rule,  we  have 

OPERATION. 

w)  ./V ((71  =  first  quotient. 
7iqi_ 
First  remainder  =  ri)  7i  (70=  second  quotient. 

Second  remainckr  =  r.2)  ri  (93  =  third  quotient. 
^273 
Third  remainder  =  0 
The  numerals  placed  at  the  bottom  of  the  letters  q  and  r, 
are  called  Subscript  JS'mnbers,  and  show  the  order  in  which 
the  quotients  and  remainders  occur. 

Letters  marked  like  the  above,  are  as  independent  as 
though  they  were  different  letters.  The  reason  why  we  use 
them  in  preference  to  different  letters,  is  because  we  can 
the  more  readily  discover  what  they  are  designed  to  repre- 
sent. 

(52.)  Now',  since  the  dividend  equals  the  divisor  multi- 
plied by  the  quotient  and  increased  by  the  remainder,  we 
have  the  following  conditions  : 

JV=5in  +  ri.  (1) 

n  =  q-iVi  +  r.2.  (2) 

ri  =  qsTo.  (3) 

Substituting  73/2  for  ri  in  (1)  and  (2),  and  they  will  be- 
come 

Y  ^=  qiTi -\- q^r^.  (4) 

n  =  q-zq^ra  +  To.  (5) 

The  right-hand  member  of  (5)  is  divisible  by  r^,  and 
therefore  its  left-hand  member  must  also  be  divisible  by  r^ ; 
that  is,  n  is  divisible  by  ro. 

6 


42  ALGEBRAIC    FRACTIONS. 

The  value  of  n,  (5),  being  substituted  in  (4),  gives 
J\r=  qiQoqsrz  +  ^ira  +  ^3^2.  (6) 

The  right-hand  member  of  (6)  will  divide  by  ro,  and 
therefore  its  left-hand  member  will  also  divide  by  ro ;  that 
is,  JV*  is  divisible  by  r-z  :  hence,  r-2  is  a  common  measure  of 
M  and  n.  It  is  also  the  greatest  common  measure.  For 
every  common  measure  of  JVand  ?i,  is  also  a  measure  of 
JY — nqi  =  ri ;  and  every  common  measure  of  n  and  ri,  is 
also  a  measure  of  n — riq-i  =  r-2.  But  the  greatest  measure 
of  r-2  is  itself.  This,  then,  is  the  greatest  common  measure 
of  ^Yandn. 

In  the  above  case  we  have  supposed  the  third  remainder 
7-3  to  =0.  Had  the  process  of  dividing  extended  still  far- 
ther, it  might  still  be  shown,  that  the  last  divisor  is  the  great- 
est common  measure  ;  hence  the  truth  of  the  above  rule. 

(53.)  It  is  obvious,  that  any  factor  common  to  but  one 
of  the  two  polynomials,  may  be  struck  out  before  dividing, 
without  affecting  the  accuracy  of  the  work. 

(54.)  Also,  either  of  the  polynomials  may  be  multiplied 
by  any  factor  before  dividing.* 

EXAMPLES. 

1.  What  is  the  greatest  common  measure  of  a'*  —  x'*,  and 
a*  -j-  a^x  —  ax~  —  x'  ? 

Arranging  the  terms  according  to  the  powers  of  cr,  and 
dividing  according  to  Rule  under  Art.  46,  we  have  for  the 


*  If  the  above  demonstration  is  deemed  too  difficult,  on  account  of  its 
making  use  of  some  of  the  principles  of  equations,  which  have  not  yet 
been  fully  explained,  the  student  must  pass  it  by,  until  he  has  gone 
through  with  the  chapter  on  simple  equations,  and  then  he  can  return 
to  it  with  pleasure  and  profit. 


ALGEBRAIC    FRACTIONS.  43 

FIRST    OPERATION. 

a^  -\-  arx  —  ax^  —  x^ 


a*  -\-  a^x  —  crx-  —  ax^ 


—  a^x-j-  u'-x"  -\-  ax^  —  x* 

—  a^x  —  a^x~  -\-  ax^  -{-  x* 

2a'X-  —  2x^  =  first  remainder. 
We  must  now  divide  o3-[-a*a;  —  aa:"  —  x'^  by  2a-x^  —  2x^; 
but  before  performing  the  division,  we  will  expunge  from 
2a'^x^  —  2x^  the  factor  2x~  (Art.  53),  which  gives  a^ — x^  for 
the  divisor  ;  hence,  we  have  for  the 

SECOND    OPERATION. 


a3 

+ 

a'x- 

-CX2  — 

-Z3 

a' 

-ax' 

a^x 



x' 

a~x 

— 

x' 

+  x 


0 
There  being  no  remainder,  the  process  must  terminate. 
The  last  divisor,  or  greatest  common  measure,  is  therefore 
a«  — x^. 

2.  What  is  the  greatest  common  measure  of  6a'  -\-  llax 
-j-  3x^  and  6a^  -f-  lax  —  3x'  ? 

In  this  example,  we  may  take  either  of  the  polynomials 
as  the  divisor,  since  they  are  each  of  the  same  degree. 

FIRST    OPERATION. 

6c^  +  llax  +  3x2l6a=  +  lax  —  3x^ 


6a-^  +    lax  —  3x- 


4ax  +  6x^  =  first  remainder. 
Before  dividing  6a^  +  lax  —  3x'  by  4flx  -(-  6x^  we  ex- 
punge from  4ax  +  6x'  the  factor  2x,  and  thus  have 


44 


ALGEBRAIC    FRACTIONS. 


SECOND    OPERATION. 


6a2-f7aa:  — 3ar 
6a2  -f  9ax 


2a  +  3x 


3a  —  X 


—  2ax  —  3x^ 

—  2ax  —  3x" 


Therefore,  2a  -j-  3a:  is  the  greatest  common  measure. 
3.  What  is  the  greatest  common  measure  of 

as  —  a%  +  Zah^  —  363  and  a^  —  bah  +  ib"-  ? 


FIRST    OPERATION. 


a3- 
a'- 

-  a'b-{- 

—  ba~h  + 

3a6^ 
4a6=^ 

—   3¥ 

4a=^6  — 
4a'6  — 

ab^ 
20a¥ 

—   363 
+  166^ 

5a6  +  465 


a +  46 


19ah"  —  1963=  first  remainder. 
Before  dividing  «"  —  bah  +  46*  by  19a6"  —  196^,  we  ex- 
punge from  this  last  polynomial  the  factor  196'. 

SECOND    OPERATION. 


a2  —  5a64-462 


a  —  6 


a  — 46 


_4a6  4-46" 
_4a5_|_4fc2 

0 
Therefore,  a  —  6  is  the  greatest  common  measure. 
4.  We  will  now  seek  the  greatest  common  measure  of 
these  polynomials  after  the  terms  have  been  arranged  ac 
cording  to  the  powers  of  6,  as  follows  : 

—  36^  +  3a6=  —  a^b  +  a'  and  46*  —  bab  -\-  a^  ? 


ALGEBRAIC    FRACTIONS.  45 

Before  dividing,  we  must  multiply  the  polynomial  —  36" 
-\-3ab' — arb -\- a^  hy  4,  in  order  that  its  left-hand  term 
may  be  divisible  by  the  left-hand  term  of  the  other  poly- 
nomial. (Art.  54.) 

FIRST    OPERATION. 

—  l2b^-{-12ab-—   4a%-i-   4ta'Mb- —  5ab  +  a^ 

—  Ub'^lbab-^—   da'b   ' — — 

I—  36  —  3n 

Multiplying  by  4, —  3a6- —     a^b -\-    4a^ 

— 12a6^—  4a26  +  16a3 

—12ab- -{- 15a^b  —    oa^ 


—  19a*6  -f-  19a^  =  first  remainder. 
Before  dividing  W  —  5ab-\-a~  by  —  IQa^fi  -f  19a\  we 
expunge  from  this  last  polynomial,  the  factor  19a-,  and  then 
dividing,  we  have  for  the 

SECOND    OPERATION. 

—    6-fa 


W  —  5ab  +  a- 
462  — 4a6 


—  46  +  0 


—  ab-\-a^ 

—  ab-\-a^ 

0 
Therefore,  — b -\- a,  or  a  —  6,  is  the  greatest  common 
measure,  same  as  before. 

4.  What  is  the  greatest  common  measure  of  the  two  poly- 
nomials   5  ^^'^^  +  l^"'^  +  ^°'^'  +  ^"'^'  -  ^"^'' 
nomiais    ^  ^20^6'^  +  38a'^6^  +  16«6^  —  106^  ? 

Ans.  3a"  +  2ab  —  6^. 

5.  What  is  the  greatest  common  measure  of 

x^  —  b'^x  and  ar^  +  26x  +  6^  ? 

Ans.  X  -\-  b. 


46  ALGEBRAIC    FRACTIONS. 

6.  What  is  the  greatest  common  measure  of 

a-  —  (ii  —  2b'  and  a^  —  Sub  +  2b~  1 

Ans.  a  —  2b. 

In  this  example  it  is  immaterial  which  polynomial  we 
consider  as  the  divisor,  since  they  are  of  the  same  degree. 

7.  What  is  the  greatest  common  divisor  of 

C  a;6  _|_  42,5  __  3^,4  _  16^3  ^  1 1^"-  _f-  i2x  —  9, 

I  %x^  +  20a;''  —  \2x^  —  48x2  ^  22a:  +  12  / 

Ans.  x^  -f~  ^'  —  ox  -\-'i. 

8.  What  is  the  greatest  common  divisor  of 

{  20z«  —  12x^  +  16x^  —  \bx^  +  14x=  —  lox  4-  4, 
J  15a;*—   9x3-f-47x=^  — 21x  +28? 

Ans.  5x=^— 3x  +  4. 


CASE   II. 

(55.)  To  reduce  a  polynomial  fraction,  that  is,  a  fraction 
of  which  the  numerator  or  denominator,  or  both,  are  poly- 
nomials, to  its  lowest  terms,  we  have  this 

RULE. 

Divide  both  numerator  and  denominator  by  their  greatest 
common  measure.,  found  by  Rule  under  Art.  50. 

EXAMPLES. 

,     r       •      36x°  — 18x^  —  27x^  +  9x3   .     .^ 

1.  Reduce  the  fraction —^^, -3-— — 77—^—  to  its 

27x"'y  —  18x*y''  —  \ix^y^ 

simplest  form. 

We  see,  by  a  mere  glance  of  the  eye,  that  the  numerator 

and  denominator  can  both  be  divided  by  9x^,  by  which  di- 

4x3  — 2x'^  — 3x+l 


vision  the  fraction  becomes 


Szaya  _  2x3/2 


ALGEBRAIC    FRACTIONS.  47 

We  must  now  seek  the  greatest  common  measure  of 

4x3  _  2x2  —  3a:  +  1  ^nd  Sary^  _  ^xy-  —  if. 
Dividing  the  second  of  these  by  y"  (Art.  53),  and  multi- 
plying the  first  by  3  (Art.  54),  we  have  the 


FIRST    OPERATION. 

12x3- 

-6x=  — 

?x  + 

33x2  — 2x  —  l 

i 

12x2- 

-8x-— 

4x 

'4x  +  2 

Multiplying  by  3, 

2x'  — 

5x  + 

3 

6x2  — 

15x  + 

9 

6x"-  — 

4x  — 

2 

—  llx-f-11  =  first  remainder. 
We  must  now  repeat  the  operation  upon  Sx^  —  2x  —  1 
and  —  llx  +  11.    Dividing  the  second  of  these  by  11  (Art. 
53),  we  have  for  the 


SECOND    OPERATION. 


3x2- 

-2x- 

-1 

-x+1 

3x2- 

-3x 

—  3x— 1 

X  — 

-1 

X  — 

-1 

0 

Hence,  the  greatest  common  measure  of  the  numerator 

4^3 2x- 3x-f-l 

denominator  of  the  fraction  ^  ^  ^ — -, — ~r-  is  — x  -\-\ 

3x2y2  —  2xy*  — 1/2 

or  X — 1.     Dividing  both  numerator  and  denominator,  of 

4x8  ^  2x 1 

the  above  fraction,  by  x —  1,  it  becomes  — - — „ for 

3xy2  -f-  2/2 

the  reduced  value  of  the  given  fraction. 

2.  Reduce  — -, — ^  to  its  lowest  terms. 

x^-[-2xT/  +  y^ 


48  ALGEBRAIC    FRACTIONS. 

In  this  example  the  greatest  common  measure  of  the  nu- 
merator and  denominator  is  found  to  be  x  -{-  y.     Hence, 


the  fraction  reduced  becomes 


x^  —  ry 


3.  Reduce ^^— —  to  its  simplest  form. 

m^  —  mrn  —  mn*  -\-n^ 

Ans.  ' —  . 

m  —  n 

4.  Reduce  — — —  to  its  simplest  form. 

Ans.  — —-. 
a  —  0 

_    ^    .         exy-\~Sx-{-   9y+12^    .      .      ,       , 

5.  Reduce  — —^-^ — ' ^^—- to  its  simplest  iorm. 

lOxy  —  Sx  +  ldy—12  ^ 

Ans  ?^- 
*5y— 4' 

^    „    ,       6x^  — 4x^  —  11x3  — 3x'  —  3.T—l       .       . 

b.  Reduce  — ; ~ ;;-- — to  its  sun- 

4X-'  -I-  2x^  —  I8x^  4-  3x  —  5 

plest  form. 

Ans. 


2xH-5 


CASE   III. 

(56.)  To  reduce  a  mixed  quantity  to  the  form  of  a  frac- 
tion. 

RULE. 

Multiply  the  entire  part  hy  the  denominator  of  the  frac- 
tion^ to  which  product  add  the  numerator^  and  under  the 
result  place  the  given  denominator. 


ALGEBRAIC    FRACTIONS.  49 


1.  Reduce  llx  -| ^^^  to  the  form  of  a  fraction. 

7x 

In  this  example  the  entire  part  is  llx,  which  multiplied 

by  the  denominator  7x,  gives  llx",  to  Avhich  adding  the 

numerator  x-\-y,  we  have  77x-  -\-  x-\-y  for  the  numerator 

of  the  fraction  sought,  under  which  placing  the  denomi- 

11  x^  —I—  X  -\-  y 

nator  Ix.  we  finally  obtain C i    for   the  reduced 

Ix 

X    I    1/ 

form  of  llx  -| !-^. 

IX 

bx  ~\~  x^ 

2.  Reduce  x to  the  the  form  of  a  fraction. 

.         mx  —  bx  —  x^ 

Ans. , 

m 

3.  Reduce  y  -{-  3x — to  the  form  of  a  fraction. 

S  -\-  a 


Ans  3v+9x  +  ay-f-3aa:  — 6 
3  -]-a 

a" 62 

4.  Reduce  x to  the  form  of  a  fraction. 

X 

Ans. — . 

X 
6 3^ 

5.  Reduce  3a^  —  6-1 to  the  form  of  a  fraction. 

1  —  y 

A      21a^  —  36  —  3aV  +  6y  —  r' 
1  —  y 

6.  Reduce  9  A to  the  form  of  a  fraction. 

a  —  x^ 

9a  — 6x=^  — 8c* 
Ans. . 


50 


ALGEBHAIC    FRACTIOXS. 


CASE   IV. 

(57.)  To  reduce  a  fraction  to  an  entire  or  mixed  quantity. 

RULE. 

Divide  the  numerator  by  the  deno/iiinator,  the  quotient 
will  be  the  entire  part ;  if  there  is  a  remainder^  place  it  over 
the  denominator  for  the  fractional  part. 


9«  —  6; 


EXAMPLES. 


to  a  mixed  quantity. 


1.  Redi 

a  —  X' 

Dividing-  the  numerator  by  the  denominator,  we  find  this 

FIRST    OPERATION. 

9fl— 6x-  — Sc^la  — x- 


9«  —  9^' 


19  =  integral  part. 

3a:"  —  Sc'*  =  numerator  of  fractional  part. 

^2"  • 


Therefore  the  quantity  sought  is  9  + 


a  —  X 

We  will  now  change  the  order  of  the  terms  of  the  nume- 
rator and  denominator,  by  placing  the  x^  first  ;  we  thus  find 
this 

SECXDND    OPERATION. 
X^  +  a 


— 6x9-h9a  — 8c* 
—  6x2  +  6a 


3a 


6  =  integral  part. 
Sc*  =  numerator  of  fractional  part 
3a  —  8c^ 


Therefore  the  quantity  sought  is  6  -j . 


ALGEBRAIC    FRACTIONS.  51 

These   two  results  are   equivalent,   but    under  illfTerent 
forms. 

2.  Reduce  to  an  entire  quantity. 

X 


Ans  a  —  X. 


3.  Reduce  — to  a  mixed  quantity, 


Ans.2x-?^±^^. 
3a: +1 

4.  Reduce  —  to  an  entire  quantity. 

in  —  y 

Ans.  )iL^  H-  my  +  ?/^ 

.     p    ,        20a*  —  10(7 -f  G  ^  .      , 

0.   Reduce to  a  mixed  quantity. 

5fl 

Ans.  4f/  — 2H . 

5a 

_    „    ,       9v'— lSy  +  8aV.  •      , 

6.  Reduce —^ — ' -^  to  a  mixed  quantity. 

yy 

Ans.  .V-  — 2+-^. 

„      r.      ,  !  t//'' 21/?   ^  .         , 

/.  Reduct to  a  nnxed  quantity. 

Ans.  2m^ —  — . 
m 


52  ALGEBRAIC    FRACTIONS. 

CASE   V. 

(58.)  To  reduce  fractions  to  a  common  denominator. 

RULE. 

'  Multiply  successively  each  numerntor  into  all  the  denomi- 
nators^ except  its  own ,,  for  new  numerators^  and  all  the  de- 
nominators together  for  a  common  denominator. 

EXAMPLES. 

1.  Reduce-,  -,  — to  equivalent  fractions  having:  a  cora- 

mon  denominator. 

a  X  2  X  7a  =  14a^  =  new  numerator  of  first  fraction. 
h  XxX  7a  =  7a6a:=new  numerator  of  second  fraction, 
f  X  .T  X  2    =2ca:  =  new  numerator  of  third  fraction. 

and  X  X  2  X  7a  =  14ca;=common  denominator. 

,_,        .        14a^       lahx      2cx  .  •     i     ^  r 

rhereiore,  :    :    are  the  equivalent  trac- 

'  14ax '    Uax '    Uax  ^ 

lions  sought. 

2.  Reduce  — ,    — ,  and  y.  to  fractions  havino;  a  common 

2a'  3x'    ^'  ^ 

denominator. 

.   9mx  4a6  6axy 
Ans.  —-  :  - —  :  -— ^. 
box  OCX   OCX 

1   x'  a^  +  x^ 

3.  "Reduce  -,  — , to  equivalent  fractions  having  a 

^    o     a  -J-  X 

common  denominator. 


3a-{-3x     2ax2-f-2x3_    6a«-f6x' 
6a  -(-  6x '     6a  -|-  6x    '     6a  -^  6x 


ALGEBRAIC    FRACTIONS.  53 

4.   Reduce  —,   — , —  to  fractions  having  a  com- 

3b     DC  a 

mon  denominator. 

5cdx      iSbdx"      15aV)c  —  l5bcx- 


Ans. 


Idbcd  '    lobcd  '  lobcd 


X    X 1     a-  +  2 

5.  Reduce-,   — - — ,    — - —  to  fractions  having  a  com- 


mon denominator. 


.        12a:      8x  — 8      6a:-+12 
^"'•"2^'    "24"'    "^4~- 


CASE    VI. 

(59.)    To  add  fractional  quantities. 

RULE. 

Reduce  the  fractions  to  a  common  denominator ;  then  add 
the  numerators,  and  jilace  their  sum  over  the  common  de- 
nominator. 

EXAMPLES. 

1.  What  is  the  sum  of  ;^,   I,   ^? 
3a    3     7 

These  fractions,  when  reduced  to  a  common  denominator, 

.  21x    21a     9ay       ,,..!.•  .  u 

become   — — ,    — — ,    — -- ;  adding  their  numerators,  we  have 
63o     63a     D3a 

21a;  4-  21a  -{-  day  ;  placing  this  over  the  common  denomi- 
nator, we  find 

X    I    1    .    y 21x-j-21a  +  9oy 7x-|-7a-|-3ay 

3^  "^  3  "^  7  63^^  ~  21^ 


54  ALGEBRAIC    FRACTIONS. 


2x  8x 

2.  What  is  the  sum  of  3x  -f-  —  and  x —  ? 

5  9 


Ans.  3x  4-  ^^  . 
45 


3.   What  is  the  sum  of—  ,    —  , 


2x     Ix     2x  +  1 
3  '     4  '    ~5~ 


^     ,  49x4-12 

Ans.  2x  A -!- — . 

^        60 


4.  What  IS  the  sum  ot  —  ,     —  ,     —  ? 
4       5       6 


,  45x  +  48x4-50x     „     ,  23x 

An>;.    1 ! =2x+— — . 

60  60 


5.  What  IS  the  sum  of  — ^ ,     — — r 


Ans. 


6.  What  IS  the  sura  of  ■ ,     :: •' 

4  4 

Ans.   "lili'. 


CASE   VII. 

(60.)  To  subtract  one  fraction  from  another. 

K,  U  L  E  . 

Reduce  the  fractions  to  a  common  denominator^  then 
auhtract  the  numerator  of  the  subtrahend  from  the  nume- 
rator of  the  minuend,  and  place  the  difference  over  the 
common  denominator. 


ALGEBRAIC    FRACTIONS.  55 


EXAMPLES. 


^         3x4- (I  2x  —  a 

1.   from —    subtract     — - — . 

4  3 

These  fractions,  when  reduced  to  a  common  denominator, 

,  9x-\-3a  Sx  —  4a 

become  -^•—   and  — -- — .     bubtraclmc:  the  numera- 

tois  we  have  9a;  +  3a  —  (8a:  —  4o)  =  j--}-7a,  placing  this 
over  the  common  denominator  12,  we  find 

3x-|-a     2x  —  0 a--f-7a 

4  3~'~~I2~  ■ 

2.  From^r-^'   subtract  !^-t_^ 
5  4 

19m  —  y 


Ans.    

20 


3.   From  3y-j--   subtract  i/ —  ^^^ ". 


c 
Ans.  2y  -j- 


4  From    — '—^  subtract   -. 

2  2 

Ans.  y 

en.  a;-  +  2a-y  -^  y"      .  i^  — 2xy  -4-  y" 

5  From    ! £_LjL    subtract  ■ ^    '    ■    . 

4x2/  4a-y 

Ans.   1 


6     From    subtract  a 


3a;  +  2i/ 

Ans.  2  —  a ' — - . 


56  ALGEBRAIC    FRACTIONS. 

CASE   VIII. 

(61.)  To  multiply  fractional  quantities  together. 

RULE. 

If  any  of  the  quantities  to  he  multiplied  are  mixed,  they 
musty  by  Case  III,  he  reduced  to  a  fractional  form ;  then 
multiply  together  all  the  numerators  for  a  numerator,  and 
all  the  denominators  together  for  a  denominator. 

EXAMPLES. 

1.  Multiply  .-p  by -:i^. 

The  product  of  the  numerators  will  be 

(x  +  c)  X{x  -\-  h)  =  X'  -{-  ax  -\-  hx  -^  ah ; 

and  the  product  of  the  denominators  is  2  X  3  =  6. 

_-.  X  A- a       x-\-h       x'^-\-ax-\-hx-\- ah 

Hence,  -t   x  ^  =  -IL^^-. 

2.  Multiply  ?lf±^yii±'. 

'^  •'      be        ''   b-\-c 

x*  —  h* 

Ans. 


hH  +  bc' 

«  3 J.     4_l_3;  3 

3.  What  is  the  continued  product  of  — -- ,  — -— ,  and  - 1 

r  7    '     2    '        7 

36  — 3x  — 3x2 

Ans. _ . 

98 

4.  What  is  the  product  of  ^^,  ^=^  ? 

,  Ans. : — . 


ALGEBRAIC    P'RACTIONS,  57 


5.  What  is  the  continued  product  of  — ,  ' ,   - ,  and 

?7l  X     '    c 

r 1  ■  .        2b/ix  —  2bdx 

^  Ans. . 

cmrx  —  cmx 

6,  What  is  the  product  of  1/  + by  ^    ,     ? 

Ans.  ^r-^y  +  r-l, 

2V- 


CASE   IX. 

(62.)  To  divide  one  fraction  by  another. 

RULE. 

If  there  are  any  mixed  quantities^  reduce  them  to  a  frac- 
tional form  ^  by  Case  III. ;  then  invert  the  divisor,  and  mul- 
tiply as  in  Case  VIIL 

EXAMPLES. 

^     ^.  ..     3x  +  7,     43:— 1 

1.   Divide   — ' —  by . 

4         •       o 

If  we  invert  the  divisor,  and  then  multiply,  we  have 

Zx  +  I  5  15^  +  35-      .  .     ^ 

—  X =  — r for  the  quotient. 

4  4a; —  1        16x  —  4 


2.  Divide  5^''- by  ^-!±l:. 


.       x'u  —  y3 
x'  +  xy- 


58  ALGEBRAIC    FRACTIONS. 

3.  Divide  -„^T-  by  ^• 

3m'      "^  5m 


4.   Divide  ^^y?^^. 
8cd     ^  4d 


.  3oa 

Ans. 


12mY 


Qc'^y  ' 


5.  Wiiat  is  the  quotient  of divided  by  -  ? 


Ans. 


a:— 1 


6.  What  is  the  quotient  of — - —  divded  by  — — ? 

Ans. . 

x—l 


SIMPLE    EQUATIONS.  59 


CHAPTER  III. 


SIMPLE   EQUATIONS. 

(63.)  Jin  equation  is  an  expression  of  two  equal  quanti- 
ties with  tlie  sign  of  equality  placed  between  them. 

The  terms  or  quantities  on  the  left-hand  side  of  the  sign 
of  equality  constitute  the  first  member  of  the  equation, 
those  on  the  right  constitute  the  second  member. 

Thus,  x  +  2  =  a,  (1) 

1-1  =  ^  (2) 

3a;  _^  7  =  c,  (3) 

are  equations  ;  the  first  is  read,  "  x  increased  by  2  equals  o." 

The  second  is  read,  "  one-half  of  x  diminished  by  1  equals 
6." 

The  third  is  read,  "  three  times  x  increased  by  7  equals 
c." 

(64.)  Nearly  all  the  operations  of  algebra  are  carried  on 
by  the  aid  of  equations.  The  relations  of  a  question  or 
problem  are  first  to  be  expressed  by  an  equation,  containing 
known  quantities  as  well  as  the  unknown  quantity.  After- 
wards we  must  make  such  transformations  upon  this  equa- 
tion as  to  bring  the  unknown  quantity  by  itself  on  one  side 
of  the  equation,  by  which  means  it  becomes  known. 


60  SIMPLE    EQUATIONS. 

(65.)  An  equation  of  the  first  degree^  or  a  simple  equa- 
tion^ is  one,  in  which  the  unknown  has  no  power  above  the 
first  degree. 

(66.)  A  quadratic  equation^  is  an  equation  of  the  second 
degree,  that  is,  the  unknown  quantity  is  involved  to  the 
second  power,  and  to  no  greater  power. 

(67.)  An  equation  of  the  third,  fourth,  &c.,  degree,  is 
one  which  contains  the  unknown  quantity  to  the  third, 
fourth,  &c.,  degree  ;  but  to  no  superior  degree. 

And  in  general,  an  equation  which  involves  the  mth  power 
of  the  unknown  quantity,  is  called  an  equation  of  the  mth 
degree. 

(68.)  The  following  axioms  will  enable  us  to  make  many 
transformations  upon  the  terms  of  an  equation  without  de- 
stroying their  equality. 


AXIOMS. 

I.  If  equal  qua^itities  he  added  to  both  members  of  an 
equation,  the  equality  of  the  members  will  not  be  destroyed. 

II.  If  equal  quantities  be  subtracted  from  both  members 
of  an  equation,  the  equality  of  the  members  will  not  be 
destroyed. 

III.  If  both  members  of  an  equation  be  multiplied  by 
the  same  quantity,  the  equality  will  not  be  destroyed. 

IV.  If  both  members  of  an  equation  be  divided  by  the 
same  quantity,  the  equality  will  not  be  destroyed. 

CLEARING  EQUATIONS  OF  FRACTIONS. 

(69.)  When  some  of  the  terms  of  an  equation  are  frac 
tional,  it  is  necessary  to  so  transform  it,  as  to  cause  the  de 
nominators  to  disappear,  which  process  is  called  clearing  of 
fractions. 


SIMPLE    EQUATIONS. 


61 


Let  it  be  required  to  clear  of  fractions,  the  following  equa- 
tion. 

Now,  by  Axiom  III,  we  can  multiply  all  the  terms  of  this 
equation  by  any  number  we  please,  without  destroying  the 
equality.  If  we  multiply  by  a  multiple  of  all  the  denomi- 
intors,  it  is  evident  they  will  disappear. 

Tf  we  choose  the  least  multiple  of  the  denominators  as  a 
multiplier,  it  is  plain  that  the  labor  of  multiplying  will  be 
the  least  possible. 

Thus,  in  the  above  example,  multiplying  all  the  terms 
of  both  sides  of  the  equation  by  G,  which  is  the  least  multi- 
ple of  2,  3,  and  6,  we  have 

3a:  +  2x  +  a;  ==  6a;  +  6.  (2) 

This  equation  is  now  free  of  fractions. 

(70.)  Hence,  to  clear  an  equation  of  fractions,  we  deduce, 
from  what  has  been  said,  this 

RULE. 

Multiply  all  the  terms  of  the  equation  by  any  multiple  oj 
(heir  denominators.  If  we  choose  the  least  common  multi- 
pie  of  the  denominators,  for  our  multiplier,  the  terms  of 
the  fraction,  when  cleared,  will  be  in  their  simplest  form. 

EXAMPLES. 

1.  Clear  of  fractions  the  equation ^  =  — — . 

^  5  2  7 

In  this  example  the  least  common  multiple  of  the  deno- 
minators 5, 2,  and  7,  is  70  ;  hence,  multiplying  all  the  terms 
of  our  equation  by  70,  we  find 

14a:  —  14a  =  35x  +  356  —  10, 
for  the  equation  when  cleared  of  fractions. 


62  SIMPLE    EQUATIONS. 

«      r^^  x- r  •  ^ ^     ,     X  4- U  X /;  .      X 

2.  Clear  ot  fractions  — 1 ]— -—  =x-\-  — 

o  4  ^  lb 

Ans.  2z  —  4  4-  4x  +  4a  —  8x  -f-  86  =  16x  -j-  x. 

(71.)  We  must  observe  that  when  a  fraction  has  the  sign 
— ,  it  requires  its  value  to  be  subtracted,  so  that,  if  it  is  writ- 
ten without  the  denominator,  all  the  signs  of  the  numerator 
must  be  changed. 

^ J       x-l-1       T 3  c 

3.  Clear  of  fractions  — \- -^ "— j— =  «  +  '' —  ;:- 

Ans.  42x-42+28a:+28— 21a:+63  =S4fl+846— 12c. 

4.  Clear  the  equation  -  +  -  +  -  +  ^  +  " =25 1  of  fractions. 

2      3     4     5      6 

Ans.  303:  +  20x  +  15a:  +  12x  -}-  lOx  =  15060. 

5.  Clear  the  equation 1 \-  -  ==  g  o{  fractions. 

Ans.  a- dm  -\-bdx  +  cmx  =  dgmx. 

^rx,         ,  •  X        ,  X  —  3       a-  —  5m. 

6.  Clear  the  equation  ; — -— r ~  =  —  ot 

^  u'—b-        a-\-b       a  —  b        a 

fractions. 

-f-  Sab  —  a"x  —  abx  -|-5a=' 


.         {  ax-\-a-x  —  abx  —  3a 
^^^-  I  -\r5ab  =  a=m  —  b-7n 


m. 


TRANSPOSITION  OF  THE  TERMS  OF  AN  EQUATION. 

(72.)  The  next  thing  to  be  attended  to,  after  clearing  the 
equation  of  fractions,  is  to  transform  it  so  that  all  the  terms 
containing  the  unknown  quantity  may  constitute  one  mem- 
ber of  the  equation. 

If  we  take  the  equation 

we  have,  when  cleared  of  fractions, 


SIMPLE    EQUATIONS.  63 

6a  —  3x  +  2bx  =  48a:.  (2) 

If  we  add  to  both  members  of  this  equation  3a;  —  2bx 
(Axiom  L),  it  becomes 

6a  —  3x  -I-  2bx  +  3a;  —  26a;  =  48a;  -j-  3x  —  26a;.      (3) 

All  the  terms  of  the  left-hand  member  cancel  each  other 
except  6a. 

Therefore  we  have 

6a  =  48r  +  3x  —  26a;,  (4) 

in  which  all  the  terms  of  the  right-hand  member  contain  x. 

If  we  compare  equation  (4)  with  (2),  we  shall  discover, 
that  the  terms  —  3a;  +  26x,  which  are  on  the  left  side  of 
equation  (2),  are  on  the  right  side  of  equation  (4),  with 
their  signs  changed. 

Hence,  we  conclude  that  the  terms  of  an  equation  may 
change  sides,  provided  they  change  signs  at  the  same  time. 

(73.)  To  transpose  a  term  from  one  side  of  an  equation 
to  the  other,  we  must  observe  this 

RULE. 

Jiny  term  may  be  transposed  from  one  side  of  an  equation 
to  the  other  J  by  changing  its  sign. 

EXAMPLES. 

X  -4-  6  5x 

1.  Clear  the  equation    -— - — f-  26  =  j  +  2  of  fractions, 

and  transpose  the  terms  so  that  all  those  containing  r  may 
constitute  the  left-hand  member. 

First,  clearing  the  above  equation  of  fractions,  by  Rule 
under  Art.  70,  we  have 

2x-f  12-fl04  =  5x-f8. 


64  SIMPLE   EQUATIONS. 

Secondly,  transposing  2a;  from  the  left  member  to  the 
right  member,  and  8  from  the  right  member  to  the  left,  we 
have    12  -j-  104  —  8  =  5a:  —  2a;  for  the  result  required. 

2.  Clear  the  equation II1_=:  7^  of  fractions,  and 

.    "-^ 

transpose  the  terms. 

Ans.  3x  —  2a;  =  45  -|-  2a. 

3.  Clear  of  fractions,  the  equation  — -3i-[-7.  =  q-|- - 

and  transpose  the  terms. 

Ans.  14a;  +  3x  — 2x  =  36  +  60. 

4.  Clear  of  fractions   and   transpose  the  terms  of  the 

X         2-\-x  c 

equation -^ . 

^  a  —  ba-\-b     a-^—b- 

Ans.  ax-\-bx  —  ax-\-bx=  c  -\-2a  —  2b. 

(74.)  We  are  now  prepared  to  find  the  value  of  the  un- 
known quantity.  If  we  take  the  last  example,  it  may  be 
written  thus, 

(a -[-6 — a  -\-b)  x:=c  -\-2a  —  2b  ; 
or  uniting  the  like  terms  within  the  parenthesis,  it  becomes 

2bx  =  c  -\-2a  —  2b. 
Dividing  both  sides  of  this  equation  by  26,  (Axiom  IV.), 

c    ,  c-l-2a  — 26 

we  nncl  x  = : 

26  ' 

hence,  the  value  of  x  is  now  known,  since  it  is  equal  to  the 

c  +  2a— 26 
expression  — ■ — — . 

(75.)  From  what  has  been  done,  we  discover  that  an 
equation  of  the  first  degree  may  be  resolved  by  the  follow- 
ing general 


SIMPLE    EQUATIONS.  65 

K  U  L  E . 

I.  If  any  of  the  terms  of  the  equation  are  fractional,  t'i> 
equation  mtist  be  cleared  of  fractions,  by  Rule  under  ^irt.  TO. 

II.  The  terms  must  then  he  xo  transposed,  that  all  Ihose 
containing  the  unknown  quantity  may  constitute  one  side  c.r 
vicinbcr  of  the  equation,  by  Rule  under  Art.  73. 

III.  Then  divide  the  algebraic  sum  of  those  terms  on  that 
side  of  the  equation  which  are  independent  of  the  unknown 
quantity,  by  the  algebraic  sum  of  the  coefficients  of  the  terms 
containing  the  unknown  quantity,  the  quotient  will  be  the 
value  of  the  unknown  quantity. 

EXAMPLES. 

X  X 

1.  What  is  the  value  of  a:  in  the  equation 1-  -  =x — 10  1 

3        4 

This,  cleared  of  fractions,  becomes 

4a;-|-3x=12a:— 120. 
When  the  terms  are  transposed  and  united,  we  have 

120  =  5x. 
Dividing  by  5,  we  get      24  =  x. 

2.  What  is  the  value  of  x  in  the  equation 

2a:  +  1       a;  4-  3  , 

X = — '■ —  ' 


Ans.  ar  =  13. 

-     _.        21— 3a;      4a:  +  6  5a:  +  1       ^    . 

3.  Given -^ —  =6 ! —  to  find  x. 

3  9  4 

Ans.  a-^=  3. 

4.  Find  x  from  the  equation  3ox -j —  —  3  =  bx  —  a. 

6  —  3fl 


66  SIMPLE    EQUATIONS. 

^     ^.        a;  —  2       3.r   ,    15x       n^   .    r    i 

5.  Given 1-  — —  =  3 / ,  to  find  x. 

4  2  2 

Ans.  a;  =  6. 

^  .   ,     ,  ...      3tx     26x       .        ,. 

6.  Find  X  so  as  to  satisty  the  condition —  4  =/• 

•^  a         m 

rt/m  +  4am 

Ans.  0:=:; — -r- 

3cw — 2a6 

„    „.    ,     ^         ,  .      Snx  —  b     36      .       ,       c 

7.  r  ind  x  trom  the  equation — 7r='* — "  —  7i- 

7  2  2 

56  4-96— 7c 
Ans.  a;  = — ■• 

167J. 

S.  Given  ^-^  +  '^-+—  =  3x-  —  12,  to  find  x. 

Ans.  a:  =6. 

^^.  3a:  —  5,   4a;  —  2  .T^ri 

9.  Given  a; — \-  — - —  =  a:  +  1,  to  find  x. 

^  Ans.  a;  =  6. 

10.  Given  ^4^  —  3x  +  ?^=^  +  3  =  a-,  to  find  x. 

3  5 

Ans.  a;  =  2. 

,       ^.        3x  — 2   ,   3a;  +  2  ^   ^    c    ^ 

11.  Given   -— 1 -^=x— l,to  finda:. 


12.  Given  -— -+x=ll,  to  find  x. 

O  I 


Ans. 


Ans.  a;^9/j. 


„    ^.         (a-{-b)x   ,        X  a;  + 1   ^    „    , 

13.  Given  ^    ^  /    +  -. r^  =  — ^,  to  find  x. 

a  —  6      'a'»  —  b^       a  +  b 


Ans.  x  = 


a^  +  2ab  +  b^  —  a-\-b-\-l 


SIMPLE   EQUATIONS.  67 

QUI^IONS,    THE    SOLUTION    OK    WHICH     KEQUIRE    EQUAT!C)NS 
OF    THE    FIRST    DEGREE. 

(76.)  In  the  solution  of  questions,  by  the  aid  of  algel)ra 
the  most  difficult  part  is  to  obtain  the  proper  equation  which 
shall  include  all  the  necessary  relations  of  the  question. 
When  once  this  equation  of  condition  is  properly  found, 
the  value  of  the  unknown  quantity  is  readily  obtained  by 
the  Rule  under  Art.  75. 

Suppose  Ave    wish  to    solve,  by  algebra,  the   lollowir 
(;uestion. 

1.  What  number  is  that,  whose   half  increased  by  i 
third  part  and  one  more  shall  equal  itself  ? 

If  we  suppose  x  to  be  the  number  sought,  its  half  will  bt 

-,  which  increased  by  its  third  part,  becomes  z  -[-  55    ^nd 

'bis  increased  by  one,  becomes   -+--}-  1,  which  by  the 

question  must  equal  itself. 

Therefore,  we  have    -  -|-     -f- 1  ^  x  for  the  equation  of 

••ondition. 

Solving  this,  by  Rule  under  Art.  75,  we  have  x  =  6, 

VERIFICATION. 

?  =  iof6=3, 

?=iof6  =  2, 
1  =1, 


Therefore,  -  -}-  ^+  1  ==  6,  which  sliows  that  6  is 


truly  the  number  sought. 


CS  SIMPLE    EQUATIONS, 

Again,  let  us  cmleavor  to  solve  tills  question  : 

2.  What  number  is  that  whose   third  part   exceeSi  its 

fourth  part  by  5  / 

Suppose  X  to  be   the  number,  then  will  its  third  part 

X       .  X 

—  :  its  fourth  part   =-. 

Therefore,  the  excess  of  its  third  part  over  its  fourth  part 

XX. 

is  expressed  by ,  which, by  the  question, must  equal  5 . 

X  X 

Hence,  we  have  the  following  equation =  5, 

3       4 

this  solvci!,  gives  x  =  60  ;  the  third  part  of  which  is  20, 

and  its  fourth  part  is  15,  so  that  its  third  part  exceeds  its 

r.iurth  part  by  5,  hence,  this  is  the  correct  number  sought. 

(77.)  The  method  of  forming  an  equation  from  the  con- 
rlitions  of  a  question,  is  of  such  a  nature  as  not  to  admit  of 
any  simple  rule,  but  must  be  in  a  measure  left  to  the  inge- 
nuity of  the  student. 

It  will  hoM'ever  be  of  assistance  to  pay  attention  to  the 
followins 


RULE 


letter^  we  must  indicate,  by  algebraic  symbols,  the  same 
operation,  as  it  would  be  necessary  to  perform  upon  the  true 
number,  in  order  to  verify  the  conditions  of  the  question. 

3.  Out  of  a  cask  of  wine  Avhich  had  leaked  aAvay  a  third 
part,  21  gallons  were  afterwards  drawn,  and  the  cask  was 
then  found  to  be  half  full :  how  much  did  it  hold  ? 

Suppose  X  to  be  the  number  of  gallons  which  the  cask 
held. 


SIMPLE    EQUATIONS.  69 

Then,  the  part  leaked  away  must  be  -. 

<j 

And  tlic  part  leaked  away,  together  with  the  quantity 

X 

drawn  off,  must  be  -  -j-  21. 
o 

Now,  by  the  question,  the  cask  is  still  half  full ;  so  that 

what  has  leaked  out,  together  with  what  has  been  drawn  off 

must  be  -. 

Hence,  we  have  this  equation,  -  =—  -|-  21, 

which,  cleared  of  fractions,  becomes,  3x  =  2a--|-  126  ; 
transposing  and  uniting  terms,  we  have  a:=  126. 

4.  There  are  two  numbers  which  are  to  each  other  as  6  to 
5,  and  whose  ditference  is  40.     What  are  the  numbers. 

Suppose  the  numbers  to  be  denoted  by  6a;  and  5x,  which 
are  obviously  as  6  to  5  for  all  values  of  x.  Now,  by  the 
question,  the  diflference  of  these  numbers  is  40.  Therefore, 
we  have  6rc  —  5a:  =  40  ;  that  is,  x  =  40. 

Hence,         6a-=  6x40  =240  ?,  ,  .        ' 

^      .^      ^^^  /  the  numbers  sought. 
5^  =  5X40=200^  ° 

5.  A  farmer  had  two  flocks  of  sheep,  each  containing  the 
same  number.  Having  sold  from  one  of  these  39,  and 
from  the  other  93,  he  finds  twice  as  many  remaining  in  the 
one  as  in  the  other.  How  many  did  each  flock  originally 
contain  1 

Suppose  the  number  in  each  flock  to  be  denoted  by  x. 

Then  the  flock  from  which  he  sold  39  will  have  remain- 
ing x— 39. 

And  the  one  from  which  he  sold  93  will  have  remaining 
X  — 93. 

Hence,  by  the  the  question,  we  have 

2x(x  — 93)=x  — 39,  or  2x— 186=x  — 39j 
transposing  and  uniting  terms,  x  =  147. 


70  SIMPLE   EQUATIONS. 

6.  Divide  the  number  36  into  three  such  parts,  that  |  of 
the  first,  -^  of  the  second,  ]  of  the  third,  shall  be  equal  to 
each  other. 

If  we  denote  the  Ihree  parts  by  2.r,  3x,  4a:,  it  is  plain  that 
■  of  the  first,  i  of  the  second,  ^  of  the  third,  will  be  equal 
for  all  values  of  x. 

Now,  by  the  question,  the  sum  of  these  three  parts  must 
equal  36. 

Therefore,  2x  +  3x  +  4x  =  36  ; 

uniting  terms,  we  have  9x  =  36  ; 

dividing  by  9,  and  we  obtain        x=^^. 

Consequently,  2a;  =2X4=    8^ 

3a:  =r  3  X  4  =  12  /  tlie  parts  sought. 
4a;  =  4x4  =  16) 

7.  Two  pieces  of  cloth  are  of  the  same  price  by  the  yard, 
but  of  different  lengths  ;  the  one  cost  $5,  the  other  $6^.  If 
each  piece  had  been  10  yards  longer,  their  lengths  would 
have  been  as  5  to  6.     What  was  the  length  of  each  piece  '\ 

Since  the  price  per  yard  was  the  same  for  both  pieces, 
their  lengths  must  have  been  to  each  other  the  same  as  the 
number  of  dollars  which  they  cost,  or  as  5  to  6' ,  or,  which 
is  the  same,  as  10  to  13. 

Therefore  we  will  denote  their  lengths  by  lOx  and  13a;. 

These  become,  when  increased  by  10, 

10a;  +  10  and  I3x  +  10, 
which,  by  the  question,  must  be  as  5  to  6. 

Hence,  6(10a;4- I0)  =  5(l3x+ 10)  ; 

or,  expanding,  60a; -f- 60  =:  65a; -j- 50  ; 
transposing  and  uniting  terms,  we  getl0  =  5x,  and  a;:=2. 
Therefore,  lOx  =10x2  =20 

13x=  13X2=26 


the  lengths  sought. 


SIMPLE    EQUATIONS.  71 

8.  Twelve  oxen  have  in  4  weeks  eaten  all  the  grass  which 
grew  on  3^  acres  of  land,  in  such  a  manner  that  they  not 
only  ate  all  the  grass  which  at  first  was  there,  but  also  that 
which  grew  during  the  time  they  were  grazing.  In  like 
manner,  have  21  oxen,  in  9  weeks,  eaten  all  the  grass  upon 
10  acres  of  land.  How  many  oxen  can,  in  this  way,  graze 
for  18  weeks  upon  24  acres  of  land  1 

Let  X  =  the  growth  in  acres  of  one  acre  of  grass  for  one 
week  ;  then  will  the  growth  of  3i  acres  for  4  weeks  equal 

3iX4Xa:=-^; 
o 

also,  the  growth  of  10  acres  for  9  weeks  will  equal 

10x9xa:  =  90x. 
Therefore,  the  whole  quantity  of  grass  eaten  in  the  first  case, 
equals 

40x_10+40.r 
^^+-3"-"~3"~- 
The  quantity  eaten  in  the  second  case  equals  10  -|-  90x. 
Hence  the  quantity  which  one  ox  eat  in  one  week  equals 
10+ 40a:     1      1        5  +  20a:  .      ,     ^ 

Again,  in  the  second  condition  the  quantity  which  one  ox 
eat  in  one  week  equals 

Now,  by  the  question,  an  ox  in  the  first  case  ate  the  same 
as  an  ox  in  the  second  case  ;  therefore  we  have 

5-f  20x  _10  +  90a:  . 

72       '~       189      ■  ^   ^ 

This,  solved,  gives  x=  —  of  an  acre. 

This  value  of  x  substituted  in  either  member  of  (1)  gives 


72  .  SIMPLE    EQUATIONS. 


—  for  the  fractional  part  of  an  acre  eaten  by  one  ex  in  one 


5^ 

54 

week ;  therefore,  the  quantity  which  1  ox  eats  in  18  weeks  is 

5  5 

—  X  18  =  —  acres. 

54  3 

Now,  the  24  acres  increasinf^  IS  weeks,  at  the  rate  of  — 

of  an  acre  for  each  acre  for  each  Avetk,  will  amount  to  CO 

acres. 

5 
Hence,        60  -H  -  =  36  oxen  for  the  answer. 

o 

9.  Divide  the  number  237  into  two  such  parts,  that  the 
one  may  be  contained  in  the  other  1:|  times.  What  are 
these  parts? 

Ans.  1051  and  131  §. 

10.  The  sum  of  $1200  is  to  be  divided  between  two  per- 
sons, A  and  B,  so  that  A's  share  may  be  to  B's  share  as 
2  to  7.     How  much  does  each  receive  ? 

Ans.  A$266|,  B$933),. 

The  above  question,  when  generalized,  becomes  like  the 
following  question. 

11.  Divide  a  number  a  into  two  such  parts,  that  the  first 

part  may  be  to  the  second  as  m  to  n.     What  are  the  parts  ? 

.  ma  na 

Ans. -, . 

m  -\-  11  ?ii  -J-  71 

12.  Divide  the  number  46  into  two  unequal  parts,  so  that 
when  the  greater  is  divided  by  7,  and  the  less  by  3,  the  quo- 
tients together  may  amount  to  10.     What  are  these  parts  ? 

Ans.  28  and  18. 

13.  In  a  company  of  266  persons,  consisting  of  officers, 
merchants,  and  students,  there  were  four  times  as  many  mer- 
chants, and  twice  as  many  officers  as  students.  How  many 
were  there  of  each  class  1 

Ans.  38  students,  152  merchants,  and  76  officers. 


SIMPLE    EQUATIONS.  1?, 

14.  Divide  the  number  a  into  three  such  parts,  thai  thi- 
second  may  be  m  times,  and  the  third  n  times  as  great  as 
the  first.     What  are  these  parts  ? 


1  -|-  Di  -f-  «     1  -}-  m  +  n     1  -\-m-\-  n 

15.  A  fiekl  of  864  square  rods  is  to  be  divided  among 
three  farmers,  A,  B,  C,  so  that  A's  part  shall  be  to  B's  as 
5  to  11,  and  C  may  receive  as  much  as  A  and  B  together. 
How  much  docs  each  receive  ? 

Ans.  A  135,  B  297,  C  432  square  rods. 

16.  Divide  the  number  a  into  three  such  parts,  that  the 
first  shall  be  to  the  second  as  m  to  ??,  and  the  second  part 
to  the  third  as  p  to  q.     What  are  these  parts  ? 

.  mpa  npa  nqa 

II I p  -\-  lip  -{-  nn     VI p  -\-  np  -j-w?'    mp  -\-  np  -\-  nq' 

17.  Divide  $1520  among  three  persons.  A,  B,  C,  so  that 
B  may  receive  $100  more  than  A  ;  and  C  §270  more  than 
B.     How  much  does  each  receive? 

Ans.  A  $350,  B  $450,  C  $720. 

IS.  A  certain  sum  of  money  is  to  be  divided  amongst 
three  persons.  A,  B,  C,  as  follows  :  A  shall  receive  $3000 
less  than  the  half  of  it,  B  $1000  less  than  the  third  part, 
and  C  is  to  receive  $800  more  than  the  fourth  part  of  tlie 
whole  sura.  What  is  the  sum  to  be  divided  7  and  what 
does  each  receive  '? 

Ans.  The  whole  sum  is  $38400.     A  receives  $16200, 
B  $11800,  C  $10400. 

19.  A  mason,  12  journeymen,  and  4  assistants,  receive 
together  $72  wages  for  a  certain  time.  The  mason  receives 
$1  dailv;  each  journeyman  \ 


74  SIMPLE   EQUATIONS. 

cents.     How  many  days  must  they  have  worked  for  this 
money  ? 

Ans.  9  days. 

20.  A  courier  who  had  started  from  a  certain  place  10 

days  ago,  is  pursued  by  another  from  the  same  place,  and 

by  the  same  way.     The  first  goes  4  miles  every  day,  the 

other  9.     How  many  days  wull  the  second  need  to  overtake 

the  first  ? 

Ans.  8  days. 

21.  A  courier  left  this  place  n  days  ago,  and  makes  a 
miles  daily.  He  is  pursued  by  another  making  h  miles  daily. 
How  many  days  will  the  second  require  to  overtake  the  first  1 

Ans. days. 

h  —  « 

22.  But,  in  what  time  will  the  second  courier  overtake 

the  first,  when  it  is  supposed  that  the  second  starts  12  days 

later  than  the  first,  and  his  speed  is  to  that  of  the  first  as  8 

is  to  3  ? 

Ans.  7}  days. 

23.  Two  bodies  start  from  the  same  place,  one  after  the 
other,  in  a  straight  line  ;  the  second  starts  n  seconds  later 
than  the  first,  and  its  speed  is  to  that  of  the  first  as  q  is  to 
p.     In  what  time  will  these  two  bodies  be  together  ? 

Ans.   -- —  seconds  after  the  setting  out  of  the  second. 
<1  —  V 

24.  Two  bodies  move  in  opposite  directions  ;  one  runs 
c  feet  in  each  second,  the  other  C  feet.  The  two  places 
from  which  they  start  at  the  same  time,  are  distant  d  feet 
from  one  another.     When  will  they  meet  1 

Ans.  y^r—  seconds. 
C-j-c 


SIMPLE    EQUATIONS.  75 

25.  But,  111  what  time  will  these  two  bodies  come  to- 
gether, when  that  which  goes  C  feet  each  second,  runs  af- 
ter the  other  ? 

Ans.   — — —  seconds. 
C  —  c 

Is  the  problem,  as  here  stated,  always  possible  ?  And 
what  is  required  for  it  to  be  possible  ?  What  does  the  ex- 
pression — signify,  when  C=cl     What  does  it  denote 

when  C<:cl 

26.  At  12  o'clock,  both  hands  of  a  clock  are  together. 
When,  and  how  often,  will  these  hands  be  together  in  the 
next  12  hours  ?  • 

'^The  hands  will  meet  11  times  ;  these  ren- 
j  centers  will  be  at  5y\  minutes  past  1,  lOyy 
"  "^"  ^minutes  after  2,  16yV  alter  3,  and  so  on,  in 
vcach  successive  hour  o/p  minutes  later. 

27.  Two  bodies  move  after  one  another  in  the  circumfe- 
r  nee  of  a  circle  which  measures  p  feet.  At  first  they  arc 
distant  from  each  other  by  an  arc  measuring  d  feet  ;  the  first 
moves  c  feet,  the  second  C  feet  in  a  second.  When  will 
these  two  bodies  be  together  for  the  first  time,  second  time, 
luid  so  on,  supposing  that  they  do  not  disturb  each  other's 
motion  1 

-'^"^-  c=-o'    &    ^^-^",&c.,  seconds. 

28.  But  when  will  they  meet,  when  the  first  begins  to 
move  t  seconds  sooner  than  the  second  ? 

29.  But  if  the  first  starts  t  seconds  later  than  the  second, 
when  will  they  meet  1 

Ans.  i=^,  e±±^,  ?£±i^',&c., seconds. 
C  —  c        C  —  c  C  —  c 


76  SIMPLE    EQUATIONS. 

80.  But  if  the  first,  instead  of  preceding  the  second,  runs 
against  it,  and  starts  from  the  same  place  t  seconds  sooner, 
when  do  they  meet  ? 

.         d  —  ct    r)-\-d  —  ct    2p-\-d  —  ct   .  , 

-^"^-    C  +  l'-C^TT'  ^— ,&-,secon,ls. 

31.  A  cistern  can  be  filled  by  three  pipes  j  by  the  first  in 
1}  hours,  by  the  second  in  3^  hours,  and  by  the  third  in  5 
hours.  In  what  time  will  this  cistern  be  filled  when  all 
three  pipes  are  open  at  once  1 

Ans.  48  minutes. 

32.  In  order  to  make  the  foregoing  problem  m.ore  general, 
let  th«  time  which  the  first  pipe  alone  takes  in  filling  the  cis- 
tern =  cr,  the  time  which  the  second  takes  in  doing  the  same 
=  6,  and  the  time  required  by  the  third  =  c.  What  ex- 
pression gives  the  time  in  which  all  three  pipes  together 
will  fill  it  ? 

.  ahc 

'  ah  -f  ac  -(-  he 

33.  A  servant  received  from  his  master  $40  wages, 
yearly,  and  a  suit  of  livery.  After  he  had  served  5  months 
he  asked  for  his  discharge,  and  received  for  this  time,  the 
livery,  together  with  $6}  in  money.  How  much  did  the 
livery  cost  1 

Ans.  $18. 

34.  A  master  hired  a  journeyman,  and  promised  him  8 
shillings  for  each  day  that  he  worked  for  him  ;  but  if  he 
worked  anywhere  elscj  then  the  journeyman  must  pay  him 
5  shillings  daily  for  his  board.  At  the  expiration  of  50 
days  they  settle,  and  the  journeyman  receives  10  pounds 
and  18  shillings.     How  many  days  has  he  worked  for  his 

master  1 

Ans.  36  days. 


SIMPLE    EQUATIONS.  77 

35.  Find  a  number  such  that  i  thereof  increased  by  i  of 
the  same,  shall  be  equal  to  l  of  it  if  increased  by  35. 

Ans.  84. 

36.  A  gentleman  spends  l  of  his  yearly  income  in  board 

and  lodging,  |  of  the  remainder  in  clothes,  and  lays  by  $200 

a  year.     What  is  his  income  .'* 

Ans.  $1800. 


EQUATIONS  OF  TWO  OR  MORE  UNKNOWN 
QUANTITIES. 

(78.)  Suppose  we  have  given  the  two  equations 

X  —  y  =  11, 

to  find  the  value  of  x  and  y. 

If  we  take  the  sum  of  the  two  equations,  we  shall  have 

2x  =  30. 
Dividing  by  2  ,  we  find 

X  =  15. 
Again,  subtracting  the  second  equation  from  the  first,  we 
have 

2y  =  S. 

Dividing  by  2,  we  obtain 

2/ =  4. 
2.  Suppose  we  nave  given  the  two  equations 

M=8,  (1) 

l  +  fe  =  ''  (^) 

to  find  the  value  of  x  and  y. 


78  SIMPLE   EQUATIONS. 

We  will  first  clear  these  equations  of  fractions,  by  mul- 
tiplying the  first  by  12  and  the  second  by  48  (Art.  70);  we 
thus  obtain 

4a:+3y=   96,  (3) 

8a:-f3.y  =  144.  (4) 

Now,  subtracting  (3)  from  (4)  we  have 
4x  =  48. 

Divided  by  4  we  find 

x=  12. 
If  we  multiply  (3)  by  2  it  becomes 

8a:  -f  6y  =  192.  (5) 

Now,  subtracting  (4)  from'  (5)  we  find 

3i/  =  48. 

Dividing  by  3  we  find 

y=16. 

3.  If  we  have  given  the  two  equations 

2x  — 3i/=   4,  (1) 

8a:  —  6y  =  40,  (2) 

to  find  X  and  y,  we  proceed  as  follows  : 

Dividing  (2)  by  2  it  becomes 

4x  — 3y  =  20.  (3) 

Subtracting  ( 1    from  (3)  we  find 

2a:=16.-.x  =  8. 

Multiplying  (1)  by  4  we  have 

8a:  —  122/  =  16.  (4) 

Subtracting  (4)  from  (2)  we  get 

63/  =  24  .•.  y  =4. 


SIMPLE   EQUATIONS.  79 


ELIMINATION    BY    ADDITION    AND    SUBTRACTION. 

(79.)  From  Avhat  has  been  clone,  we  discover  that  an 
unknown  quantity  may  be  eliminated  from  two  equations, 
by  the  following 


RULE. 

Operate  upon  the  two  given  equations^  by  multiplication 
or  division,  so  that  the  coefficients  of  the  quantity  to  he 
eliminated  may  become  the  same  in  both  equations;  then  add 
or  subtract  the  tu-o  equations,  as  may  be  necessary,  to  cause 
these  two  terms  to  disappear. 

EXAMPLES. 


4.  Given,  to  find  x  and  y,  the  two  equc 

itions 

3x—   y=3, 

(1) 

2/  +  2x  =  7. 

(2) 

If  we  add  the  two  equations,  we  have 

5x=l0.-.x  =  2. 

(3) 

Again,  multiplying(3)  by  2,  we  get 

2x  =  4. 

(4) 

Subtracting  (4)  from  (2)  we  obtain 

7/  =  3. 

5.  Given,  to  find  x  and  y,  the  two  equations 

-+•-=6, 
2^3         ' 

(1) 

3^2 

(2) 

80  SIMPLE    EQUATIONS. 

Clearing   these    equations  of  fractions,  by    inultlplyiiu 
each  by  6,  they  become 

3a:  +  2y  =  36,  (3) 

,2x  +  3y=39.  (4) 

Multiplying  (3)  by  3,  and  (4)  by  2,  they  become 


9it:H-6y=108, 

(5) 

4x  +  6y=   78. 

(6) 

Subtracting  (6)  from  (5)  ^ve  get 

5x  =  30, 

and           x=6. 

(') 

Multiplying  (7)  by  2,  it  becomes 

2a:=12. 

{^) 

Subtracting  (8)  from  (4),  we  find 

3?/ =  27  .-.^  =  9. 

6.  Suppose  we  wish  to  find  x,  y,  and  z,  from  the  three 
equations 

5x— 62/  +  4c  =  15,  (1) 

7x+4y-3c  =  19,  (2) 

2a:  +    y  +  6z=46.  (3) 

We  will  first  eliminate  y:  for  this  purpose  multiply  (3), 
first  by  4  and  then  by  6,  and  it  will  give 

8a: +  4y +  242=384,  (4) 

12a:  +  6y  4- 36r  =  276.  (5) 

Add  (1)  to  (5)  ;  and  subtract  (2)  from  (4),  and  we  have 
17x-f  40-r=291,  (6) 

x  + 27c  =165.  (7) 

We  have  now  the  two  equations  (6)  and  (7),  and  but 
two  unknown  quantities  x  and  c. 

Multiply  (7)  by  17  and  it  will  become 

17x+459z  =  2805.  (8) 


SIMPLL    EQUATIONS.  81 

Subtracting  (6)  from  (8)  we  obtain 

419c  =  2514.  (9) 

Dividing  (9)  by  419,  we  find 

c  =  6.  (10) 

Multiplying  (10)  by  27,  we  find 

27z=162.  ;il) 

Subtracting  (11)  from  (7)  we  get 

x  =  3.  (12) 

Multiplying  (10)  by  6,  and  (12)  by  2,  and  then  taking 
their  sum,  we  find 

6c  +  2.T  =  42.  (13) 

Subtracting  (13)  from  (3),  we  get 

(80.)  We  will  now  repeat  the  solution  of  this  last  ques- 
tion, adopting  a  simple  and  easy  method  of  indicating  the 
successive  steps  in  the  operations. 

The  method  which  we  propose  to  make  use  of,  is  to  indi- 
cate by  algebraic  signs,  the  same  operations  upon  the  respec- 
tive numbers  of  the  different  equations,  as  we  wish  to  have 
performed  upon  the  equations  themselves. 

Thus, 

(Q\  ^(A\y,oS  shows,  that  equation  (6)  is  obtained  by 
(  multiplyino;  equation  (4)  by  3. 


iplying  equation  (4)  by 
Jshf 
Ihy 

(ll)^(6)-(3) 


/,QN /r-x  I  /-.N  ^  shows,  that  equation  (10)  is  obtained 

V     ^  —  '^ '  ^  +  ^   ^  ;  by  adding  equations  (7)  and  (1). 


shows,  that  equation  (11)  is  obtained 
by  subtracting  equation  (3)  from  (G). 

/,c\ /-|4^x  _^<^  S  shows,  that  equation  (15)  is  obtained 

(  by  dividing  equation  (14)  by  3. 

And  so  on  for  other  combinations. 
11 


82  STMPI.E    EQUATIONS. 

This  kind  of  notation  will  become  familiar  by  a  little 
practice. 

We  will  now  resume  our  equations  of  last  example. 

r  Sx  — 6?/  +  4=rrzl5  (1)^ 

Given  <  7x  4-  4y  —  3c  =  19  (2)  >  ,  to  find  x,  y,  and  z. 

(2x+    y  +  6z  =  4.6  (3)) 

8x--f   4y+  24^  =  184  (4)=:(3)x4 

12a:  4-    6y-|-   36z==276  (5)r=(3)x6 

17x+   40c  =  291  (6)  =  (l)+(5) 

a: +  27-    =165  (7)  =  (4)— (2) 

17a:  4-459r=  2805  (8)  =  (7)xl7 

419c  =  2514  (9)  =  (8)— (6) 

c  =  6  (10)  =  (9)^419 

27z  =  162  (11)  =  (10)  X  27 

a:  =  3  (12)=(7)-(11) 

62  =  36  (13)  =  (10)X6 

2x  =  6  (14)  =  (12)X2 

6c-f-2a:  =  42  (15)  =  (13)  +  (14) 

y=4  (16)  =  (3)-(15) 

Collecting  equations  (12),  (16),  and  (10),w^e  have 

ra:  =  3.  (12) 

Ans.  <  1/  =  4.  (16) 

(z  =  6.  (10) 

We  will  solve  one  more  set  of  equations  by  this  method, 
giving  all  the  steps  at  length,  the  better  to  illustrate  this 
notation. 


Given* 


7a:  — 22  +  3w=17 

(1)^ 

4y  —  2z+    f=n 

(2)/ 

5y__3a:_2«=    8 

(3)> 

,  to  find  a:,  y,  z,  u,  t 

4y^Su  +  2t  =    9 

(^)\ 

3z-f.8M  =  33 

(5)^ 

SIMPLE    EQUATIONS.  83 

8y  — 42-f2^  =  22  (G)  =  (2)  X  2 

4y  —  4z-\-3u  =  l3  (7)  =  (6)  — (4) 

21x  — 6z-t-9M  =  51  (8)  =  (1)  X  3 

35y— 21x— 14w  =  56  (9)  =  (3)  X  7 

35y  — 6:  — 5w=107  (10)  =  (8)  +  (9) 

140y—140z -1-105/^=455  (11)  =  (7)  X  35 

140y— 24c— 20/^=428  (12)  =  (10)  X  4 

—116c -1-125/^  =  27  (13)  =  (11)— (12) 

348c -f  928/^  =  3828  (14)  =  (5)  X  116 

—348c  -h  375?^  =  81  (15)  =  (13)  X  3 

1303/i  =  3909  (16)  =  (14) -f  (15) 

«  =  3  (17)  =  (16)  -^  1303 

8u  =  24  (18)  =  (17)  X  8 

3c  =  9  (19)  =  (5)  — (IS) 

c  =  3  (20)  =  (19)  ^  3 

3«  =  9  (21)  =  (17)  X  3 

4c  =  12  (22)  =  (20)  X  4 

4c  — 3/<  =  3  (23)  =  (22)  —  (21) 

43/ =16  (24)  =  (23)  +  (7) 

y  =  4  (25)  =  (24)-^4 

8y  =  32  (26)  =  (24)  X  2 

Sy  —  4c  =  20  (27)  =  (26)  —  (22) 

2^  =  2  (28)  =  (6)  —  (27) 

t  =  1  (29)  =  (28)  -^  2 

2c  =  6  (30)  =  (20)  X  2 

3u  — 2c  =  3  (31)  =  (21)  —  (30) 

7x=14  (32)  =  (1)  —  (31) 

x  =  2  (33)  =  (32)-^7 
Collecting  equations  (33),  (25),  (20),  (17),  (29),  we  have 

'x  =  2. 


84  SIMPLE    EQUATIONS. 


ELIMINATION    BY    COMPARISON. 

(81.)  We  may  a. so  eliminate  one  of  the  unknown  quan- 
tities of  two  equations,  by  the  following  process  : 
Take  the  two  equations 

5y  — 4a:  =  — 22,  (1) 

4y  +  4a:==38.  (2) 

If  we,  for  a  moment,  consider  y  as  a  known  quantity, 
we  may  then,  from  each  of  these  equations,  find  the  value 
of  X  by  Rule  under  Art.  75. 
We  thus  find 

22  +5y 


4 


(3) 


38-31/ 
X  ^  — ^— .  (4) 

Putting  these  two  values  of  x  equal  to  each  other,  we  have 
22+5y       38 -By 

-^-  -  —T-  ■  ^^^ 

Clearing  (5)  of  fractions,  it  becomes 

22  +  52/  =  38  —  3y,  (6) 

transposing  and  uniting  terms,  we  find 
Sy  =  16  .-.  y  =  2. 
This  value  of  y  substituted   in  either   of  the  equations  (3) 
or  (4),  will  give 

x=8. 
The  above  method  of  eliminating  may  be  given  as  in 
the  following 

RULE. 

I.  Find,  from  each  of  the  given  equations,  the  value  oj 
one  of  the  unJaiown  qttantities,  by  Rule  under  Art.  75.,  on 
the  supposition  that  the  other  qaantities  are  knovm. 


SIMPLE    EQUATIONS.  85 

II.  Then  equate  these  different  expressions  of  the  value  of 
the  tinknown,  thus  found  ^  and  we  shall  thus  have  a  number 
of  equatiojis  one  less  than  were  first  given;  and  they  will 
also  contain  a  numher  of  unknown  quantities  one  less  than 
at  first. 

III.  Operating  with  these  new  equations  as  was  done  with 
the  given  equations,  we  can  again  reduce  their  number  one; 
and  continuing  this  process  we  shall  finally  have  but  one 
equation  containing  but  one  unknown  quantity,  which  will 
then  become  known. 

EXAMPLES. 

(7x  +  5y  +  2c=    79     (1)) 
1 .  Given  I  8a;  +  7y  +  92  =  122     (2)  \  ,  to  find  x,  y, 
t    x+4y-^oz=    55     (3)  !l      and  c. 

By  Rule  under  Art.  75,  we  find,  by  using  (1),  (2)  and  (3), 
_19—5y  —  2z 


7 


(4) 


122^^.^        (5) 


a;  =  55  — 4y— 5-.        (6) 

Equating  (4)  and  (6)  ;  and  (5)  and  (6),  we  have 

79  — 5y  — 2z   ^.   , 

Y =5o—4y—5z,        (7) 

122  — 7y  — 9z   „   .    .    ,_, 
^ =o5—4y  —  oz.      (8) 

When  cleared  of  fractions,  (7)  and  (8)  become 

79  _  5y  _  2r  =  385  —  28y  —  35z, 
122  —  ly  —  9z  =  440  —  32y  —  40r. 

Transposing  and  uniting  terms,  we  have 

23y  +  33r  =  306,  (9) 

25y-j-31c=318.  (10) 


86  SIMPLE    EQUATIONS. 

Equations  (9)  and  (10)  give 

_306  — 33z 
y~        23        ' 


(11) 


318  — 31z 


Equating  (11)  and  (12),  we  have 

306— 33z     318  — 31z 


(13) 


23  25         ' 

which  reduced  gives 

c  =  3. 
This  value  of  z  substituted  in  (11)  gives 

2/  =  9. 
And  these  values  of  z  and  y,  substituted  in  (6),  give 


(  ^^4-^2/+^-  =  62^ 
2.  Given  ?  ^x  +  jy -f-  ^2  =47  >  j  ^o  fi"^  ^5  V)  ^^^  2; 
l\x-^\y  +  lz=ZsS 

These  equations,  when  cleared  of  fractions,  become 

6.T+    4y+    2,z=    744,  (1) 

20x+15y+12z  =  2820,  (^) 

15a:  +  12y  -j-  10-  =  2280.  (3) 

Til 

From  (1),  (2),  and  (3),  we  find 

_744  — 6a:  — 4i/ 


"        3 
2820  — 20a:— 15y 

"12  ' 

2280— 15a:— 12y 


(4) 

(5) 
(6) 


10 

Equating  (4)  with  (5),  and  (4),  with  (6),  we  have 


SIMPLE    KQUATIONS. 

87 

744  —  6x  —  \y       2820  —  20x  —  15y 
3                                 12               ' 

(') 

744  —  6x  —  4y      2280  —  15a;  —  \2y 
3                                10 

(8) 

Equations  (7)  and  (8)  when  reduced  become 

4a;  +    y=156, 

(9) 

15a;  +  4y  =  600. 

(10) 

Equations  (9)  and  (10)  give 

y=156—   4x, 

(11) 

600  —  15x 

y-    4    • 

(12) 

Equating  (11)  and  (12) ,  we  have 

,_       ,         600  — 15a; 

lo6  —  4a;  = . 

4 

(13) 

This  reduced,  gives 

a;  =  24. 
Having  found  z,  we  readily  find  y  and  z  to  be 
y=60;   ^  =  120. 

ELIMINATION    BY    SUBSTITUTION. 

(82.)  There  is  still  another  method  of  elimination. 
1.  Suppose  we  have  given  the  two  equations 


5x  +  2y  =  45, 

(1) 

4a;  +    3/ =  33. 

(2) 

From  the  first  we  find 

45— 5x 

y=      2    • 

(3) 

Substituting  this  value  of  y  in  (2),  we  have 

,  45 — 5a;       „„ 
4x  -j =  33. 

(4) 

88  SIMPLE    EQUATIONS. 

Equation  (4),  when  cleared  of  iVaclions,  becomes 

8x-f-45 — 5a:  =  66.  (5) 

This  gives 

a:  =  7. 
Substituting  this  value  of  a:  in  (3),  we  find 
3/  =  5. 
2.  Again,  suppose  we  have  given,  to  find  x,  i/,  and  z,  the 
three  equations 

2x-{-4y  —  3z  =  22,  (1) 

4.x  —  2y-\-5z  =  18,  (2) 

6x  +  ly—    z  =  63.  (3) 

From  equation  (3)  we  obtain 

c  =  6a;  +  7y  — 63.  (4) 

Substituting  this  value  of  c,  in  (1)  and  (2),  and  they  will 
become 

2x -|- 42/ —  3(6a:  4- 73/ —  63)  =  22,  (5) 

4a:  — 21/ -f- 5(6x4- 71/  — 63)  =18.  (6) 

Equations  (5)  and  (6)  become,  after  expanding,  transpo- 
sing, and  uniting  terms, 

l6a:-(-l''y=167,  (7) 

34a:  -\-  33y  =  333.  (8) 

Equation  (7)  gives 

This  value  of  a:,  substituted  in  (8),  gives 

34067-17^)^^33^^333  ^^^^ 

Equation  (10),  when  solved  as  a  simple  equation  of  one  un 
known  quantity,  gives 

Substituting  this  value  of  y  in  (9),  w^e  find 

z  =  3. 
Using  these  values  of  x  and  y  in  (4),  we  obtain 

1  =  4. 


SIMPLE    EQUATIONS.  89 

(83.)  This  method  of  climiniiting-  may  be  comprehended 
in  the  followintr 


RULE. 

Having  found  the  value  of  one  of  the  unknown  quantities^ 
from  either  of  the  given  equations^  in  terms  of  the  other 
unknown  quantities,  substitute  it  for  that  unknown  quantity 
in  the  remaining  equations,  and  we  shall  thus  obtaiji  a  new 
system  of  equations  one  less  in  number  than  those  given. 
Operate  with  these  new  equations  as  with  the  first,  and  so 
continue  until  we  find  one  single  equation  with  hut  one  un- 
known quantity,  ichich  will  then  become  known. 

EXAMPLES. 

/  X  —  u'=  50  {l)\ 
,  r.-  \'iy  —  x=  120  (2)( 
1.  Criven  ^2- _  y  ^  120         ^3)?' ,  to  findw,  x,  y,  and  z. 

(3iy  — -  =  195         (4)) 

From  (1)  we  find 

w  =  X  —  50.  (5) 

This  value  of  ic,  substituted  in  (4),  gives 

3(j;_50)— -  =  195,  or  3x— r  =  345.       (6) 
Equation  (6)  gives 

z=3x  — 345.  (7) 

This  value  of  z,  substituted  in  (3),  gives 

2(3x  — 345)— 2/=  120,  orGo:— y  =  8l0     (8) 
Equation  (8)  gives 

y  =  6x  —  SlO.  (9) 

This  value  of  y,  substituted  in  (2),  gives 

3(6x— 810)  — 2:  =  120,  (10) 

or  17a:  =  2550.  (11) 

.•.a:=150.  (12) 

12 


90  SIMPLE    EQUATIONS. 

This  value  of  x  causes  (9)  to  become 

3/  =  90. 
Using  the  value  of  x  in  (7),  we  find 

z=105. 
Finally,  using  the  value  of  x  in  (5),  we  find 
w  =  100. 
rx  +  ly  =  a,  {\)\ 

2.  Given  }y-{-\z  =  a^  (2)  >  to  find  a:,  y,  and  z. 

{z-\r\x  =  a,  (3)) 

Equation  (3)  gives 

•      ■  ■-=—■  w 

This  value  of  c,  substituted  in  (2) ,  we  have 

I   4a  —  X  ,_. 

y+-^^=«-  (5) 

Clearing  of  fractions  and  uniting  terms,  (5)  becomes 

12y  —  x  =  8a.  (6) 

From  (6)  we  find 

x  =  12y  —  Sa.  (7) 

This  value  of  a:,  substituted  in  (1) ,  gives 

122/ -  8a +1  =  a.  (§) 

Equation  (8)  gives 

25y=18a,  (9) 

Therefore,  y  =  — r-- 

'  ^        25 

This  valup.  of  y,  substituted  in  (7),  gives 

_  16a 
^~  25  * 
Substituting  for  x,  in  (4),  its  value  just  found,  we  have 

^_21a 

''~25' 


SIMPLE    EQUATIONS.  91 

Hence,  collecting  values,  we  have 


We  may  observe  that  if  a  is  any  multiple  of  25,  the  above 
values  of  x,  y,  and  z  will  be  integers. 

(84.)  All  equations  of  the  first  degree,  containing  any 
number  of  unknown  quantities,  can  be  solved  by  either  ot 
the  Rules  under  Articles  79,  81,  and  83,  or  by  a  combina- 
tion of  the  same. 

The  student  must  exercise  his  own  judgment,  as  to  the 
choice  of  the  above  Rules.  In  very  many  cases  he  will  dis- 
cover many  short  processes,  which  depend  upon  the  parti- 
cular equations  given. 

(85.)  We  will  now  solve  a  few  equations,  and  shall  en- 
deavor to  effect  their  solution  in  the  simplest  manner  possi- 
ble. 

,     ^.         ^  6x -I- 53/ =  128,  L    .    1,,        1         ^         , 

1.  Given  \  „  ^^' >  to  find  the  values  of  a;  and  ■?/. 

nx  +  43/  ==    88,  )  ^ 

Adding  the  two  equations,  and  dividing  the  sum  by  9,  we 
find 

x  +  y  =  2i.  (1) 

Multiplying  (1)  by  3,  and  subtracting  the  result  from  the 
second  of  the  given  equations,  we  have 

i/=16.  (2) 

Subtracting  (2)  from  (1),  we  get 
x=8. 

2.  Given  ^  y  +  r  =  6,     (2)  >  to  find  x,  y,  and  2. 
■C:      (3)) 


92  SIMPLE    EQUATIONS. 

Dividing  the  sum  of  these  three  equations  by  2,  we  find 
.+y  +  .  =  "-+A±i.  (4) 

From  (4)  subtracting,  successively,  (2),  (3),  and  (1),  we 
find 

b  +  c-h 

2        ' 

a  —  c  +  b 


2—'    r  W 

a-\-b-{-c 


Equations  (1),  (2),  and  (3),  of  this  last  example,  are 
so  related  that  if  in  (1)  we  change  x  to  y,  y  to  z,  and  a 
to  b,  it  will  correspond  with  (2).  Again,  if  in  (2)  we 
change  y  to  z,  z  to  x  and  b  to  c,  it  will  correspond  with  (3). 
Also,  if  in  (3)  we  change  z  to  x,  x  to  y,  and  c  to  a,  it  will 
give  (1),  from  which  we  first  started.  In  each  change  we 
have  advanced  the  letters  one  place  lower  in  the  alphabeti- 
cal scale,  observing  that  when  we  wish  to  change  the  last 
letters  of  the  series,  as  z  or  c,  we  must  change  them  respec- 
tively to  X  and  a,  the  first  of  the  series. 

Since  the  above  changes  can  be  made  with  the  primitive 
equations  (1),  (2),  (3),  without  altering  the  conditions  of 
the  question,  it  follows  that  the  same  changes  can  be  made 
in  any  of  the  equations  derived  from  those.  Thus,  execu- 
ting those  changes  in  equations  (A),  we  find  that  the  first 
is  changed  into  the  second,  the  second  into  the  third,  and 
the  third  in  turn  is  changed  into  the  first. 


SlMn-E    EQUATIONS.  93 


Vx       y 

(1) 

3.  GivenJl  +  l_A 

\y     ~ 

(2) 

(>\-' 

^3) 

If  we  take  the  sum  of  these  three  equations,  we  shall  ob- 
tain 

0  0  9 

-  +  ^  +  ^---^'  +  ft  +  c.  (4) 

Now,  subtracting  twice  (2)  from  (4),  and  we  have 

^-  =  a-Y  c  —  b.  (5) 

In  a  similar  manner  subtracting  twice  (3)  and  (1),  suc- 
cessively, from  (4),  and  we  find 


?  =  a-o+i; 

(6) 

2 

-  =  —  a  +  b  +  c. 

i?) 

Equations  (5),  (6),  and  (7),  readily 

give 

2               ^ 

2              V 

(B) 

2            \ 

• 

-a-^b  +  c   J 

The  letters  in  this  example  will  admit  of  the  same  changes 
as  those  pointed  out  in  the  last  example.  Indeed,  the  only 
difference  between  the  two  examples  is,  that  the  unknown 
quantities  in  the  one  example  are  the  reciprocals  of  those  in 
the  other.     Consequently  the  expressions  for  a*,  y^  and  c, 


94  SIMPLE    EQUATIONS. 

as  given  by  equations  (B),  ought  to  be  the  reciprocals  of 
those  given  by  equations  (A),  which  we  find  to  be  really  the 
case. 

4.  Given  ^  3/+6(x+z)=7j,        (2)  >  to  find  x,  y,  and  z. 
(z-\-c{x-^y)-'--p,        (3)) 

If  we  add  and  subtract  ax  from  the  left-hand  member  of  (1), 
and  add  and  subtract  by  from  the  left-hand  member  of  (2), 
and  add  and  subtract  cz  from  the  left-hand  member  of  (3), 
they  will  become 

{l-a)xi-a{x  +  y-\-z)=m,  (4) 

{l-b)y-\-b{x-]-y-{-z)  =  n,  (5) 

{l-c)z-{-c{x  +  y-{.z)=p.  (6) 

If  we  divide  (4)  by  1  —  a,  and  (5)  by  1 — b,  and  (6)  by 
1  —  c,  they  will  become 

x+--^(x  +  y  +  z)=-^;  (7) 

1  — a  1  —  a 

'         ^    .(a:  +  y4-z)=-^:  (8) 


Taking  the  sum  of  (7),  (8),  and  (9),  we  have 
^    _i       ^       I      P 


(10) 


Therefore,  x-j- 2/ -f- 2^ f .      ^^^^ 


1  +  ,-^+ 


I  — a  '    1—6 


SIMPLE    EQT^ATIONS.  96 

This  value  of  x-j-y  +  2  substituted  in  (7),  (8),  and  (9), 
gives 


(      _^ [_      ^       \       P 


1  — a        1  —  a).,a  b  c 


'-,  (12) 


1— a   '    1—b   '    1-T 


n 


-       .      l—a~^  l—b~^   1  —  c 
1  — a       1  —  b        1  —  c 


._      P 


>      1— g.      1  — 6"^1  —  c 


(14) 


1  —  a   '    ]—b       1  —  c 


This  example  affords  a  l:eautiful  illustration  of  the  law 
of  permutations  which  can  be  made  with  the  letters  which 
enter  into  symmetrical  equations.  The  primitive  equations 
(1),  (2),  (3);  the  three  equations  (4),  (5),  and  (G) ;  and 
the  three  (7),  (8), and  (9) ;  as  well  as  the  three  (12),  (13)  and 
(14),  can  be  deduced  in  succession  from  each  other,  by 
simply  advancing  the  letters  one  place  lower  in  the  alpha- 
betical scale.  Equations  (10)  and  (11),  which  contain  all 
the  different  letters,  are  of  such  a  form  as  not  to  change  by 
this  method  of  permuting.  Consequently  the  expression 
within  the  braces  of  (12),  (13),  (14),  which  is  the  right- 
hand  member  of  (11),  must  remain  unchanged  for  the  values 
ofx,  y,  andc.  By  studying  carefully  the  different  laws 
by  which  changes  may  be  made,  we  have  great  control 
over  symmetrical  algebraic  expressions  which  we  could  not 
otherwise  obtain.  It  is  not  always  necessary  that  the  change 
should  be  in  alphabetical  order,  but  may  vary  according  to 


96  SIMPLE   EQUATIONS. 

any  other  law.  The  principle  may  be  thus  stated  :  what- 
ever changes  can  be  made  among  the  letters  entering  into 
the  primitive  equations,  without  altering  the  equations,  the 
same  changes  may  be  made  on  any  of  the  derived  equa- 
tions. 

This  method  of  deducing  one  expression  from  ano- 
ther of  a  similar  nature,  is  of  great*  use,  especially  in  the 
higher  parts  of  analysis.  In  order  that  the  proper  permu- 
tations may  be  made  with  ease,  and  without  danger  of  error, 
we  must  adopt  some  simple  and  uniform  notation  for  the 
different  values  of  the  quantities  which  enter  into  our  ex- 
pressions. Indeed,  by  a  well  chosen  method  of  notation, 
we  may  frequently  resolve,  with  ease,  questions  which 
would  otherwise  be  extremely  difficult. 

Perhaps  we  can  not  better  impress  upou  the  student,  the 
importance  of  a  judicious  notation,  than  by  giving,  at  length, 
the  solution  of  the  two  following  questions. 

5.  Find  7i  numbers,  such  that  the  first  increased  by  Ci 
times  the  sum  of  all  the  others,  shall  equal  61 ;  the  second, 
increased  by  og  times  the  sum  of  the  others,  equals  62  ;  the 
third,  increased  by  03  times  the  sum  of  the  rest,  equals  ^3  ; 
and  so  on  for  the  other  numbers. 


Let  the  n  numbers  sought  be  represented  by 
Xi,  Xa,  X3,     ------     x„. 

Then,  if 

S  =  xi+X2-|-X3  4-     -         -         -         -     -\-x„,     (a) 

we  shall  have,  by  the  conditions,  the  following  system  of 

equations  : 


SIMPLE    EQUATIONS. 


97 


3^1  +  fll(S  — Xi)=6i,  (1) 

xo  +  a,{S-x.)=^h,,  (2) 

x;  +  as{S—x,)=b,,  (3)( 


(A) 


Xn-{-an{S-x„)=b..  {n). 

From  (A)  we  readily  find  the  following  system  of  equa- 
tions : 


"■     xs-    ^' 


X3- 


01—  1 

(1-2 
flo—  1 

03 1 


«,  — r 


X  S— 
X  5— 


b. 

Oo—  i' 
as— 1' 


(1': 

(2') 

(3')>     (A') 


:r„=-i^x5-         ^" 


On—  I 


««—  1 


(^') 


Taking  the  sum  of  the  n  equations  (A') ,  we  find 

S=B' X  S—B".  (B) 

Where,  for  the  sake  of  brevity,  we  have  put 

«i  — l^Go— 1^03— 1^  ^a,—  !'"^^ 


D"=-^+      & 


ai—1    '    ao—l    '   03—1"^  ^a„— 1'    ^°   ^ 


Returning  to  equation  (B) ,  we  find 


D'  — 1 
This  value  of  5  written  in  the  n  equations  (A')  gives 


(C) 


13 


98  SIMPLE    EQUATIONS. 


X2 


D'— 1    ^       «,— 1 


0.  —  1 


(D) 


D" 


If?/ =10;  and  bi=bo  =  l.3=bi  =  b5=b^  =  bj==h=h 
=  6io=845693;    and    Oi=i,    «2  =  i-,   03=!)    04=ij 

will  the  above  question  agree  with  one  in  the  Higher  Arith- 
metic, which  question  is  there  required  to  be  solved  by  rules 
purely  arithmetical.  The  preceding  question,  is  also  a 
particular  case  of  the  above  question. 

G.  Suppose  n  individuals,  Ai,  Aa,  A3,  .  -  -  -  A„, 
play  together  on  this  condition,  that  the  one  who  loses  shall 
give  to  each  of  the  others  as  much  as  they  then  have. 
First  Ai  loses,  then  As,  then  A3,  then  A4,  and  so  on,  until, 
in  turn,  they  have  all  lost ;  and  at  the  end  of  the^ith  game 

their  respective  shares  are  Oj,  oo,  035 "n-     How 

much  had  each  before  playing  ? 

SOLUTION. 

Let  their  respective  shares  before  playing  be  represented  by 

X],  X2.X3J  -  -  -  -  x„. 

Also,  put 

a:, +X0  +  X3+     -----     +Xn  =  S.       (1) 
Since  Ai  loses  on  the  first  game,  he  must,  by  the  question, 
give  to  Aa,  A3,  A4,  &c.,  as  much  as  they  now  have.     Hence 


SIMPLE    EQUATIONS.  99 

Ai's  money  will  be  diminished  by  Xo  -|-  X3  -|-  x.|  H \-  x„, 

which,  by  (1),  equals  5  —  Xi,  so  that  Ai's  money  will  be 

Xi  —  {S  —  Xi)=2xi—S. 
Therefore,  at  the  end  of  the  first  game,  they  will  have 

Ai,           A.J,           A3,  A„, 

2  Ji  —  S ;    2xo ;  2x3 ; 2x„ . 

Now,  since  A2  loses  on  the  second  game,  he  must  give  to 
Ai,  A3,  A4,  &c.,  as  much  as  they  now  have.  Hence  Aa's 
money  will  be  diminished  by 

2x1  —  5' 4- 2x3  +2x4  + 2x„.  (2) 

Since  they,  all  together,  always  have  the  same  amount  as 
at  first,  we  have 

2x1  —  5  4-  2x,  +  2x3  H 2xn=S;  (3) 

.-. 2xi  —  iS  +  2x3  +  2x4  H 2x„  =  S—2X'2. 

Hence,  Ao,  after  the  second  game,  will  have 

2xo  —{S  —  2x,)  =  4x0  —  S. 
Therefore,  at  the  end  of  the  second  game,  they  will  have 
Ai,            Ae,            A3,            A4,  A„, 

4xi  —  2  .S;  4x2 — S;     4x3;  4x4; 4x„. 

Proceeding  in  this  way,  we  find  that  after  the  third  game, 
they  will  have 

Ai,  Ao,  A3,  A4         As,  A„, 

Sxi— 4  S;  8x2  —  2  S;  8x3  —  S;  8x4 ;       8x5;  -  -  -  8x„. 

And  in  general,  after  the  7ith  game,  they  will  have 

Ai,  A2,  A3,  A„, 

2"^^  —  2"-'S;  2"x.2  —  2"-'S;  2"Xi  —  2"-^  5;  -  -  -  2''x„  —  S. 

Equating  these  results  with  the  values, 

fil,    02,    03}    04} ^"1 


100 


SIMPLK    EQUATIONS. 


we  have  the  n  followin;^  equations 
2"Xi  —  2"-'S=ai^ 

2"a:3— 2"-^S=fl3, 


(1') 
(2') 

(4': 


(A) 


2"x„_i  — 2  6'=a„_i,     (("-!)') 
From  tl)c  above  system  of  equations  (A),  we  readily  find 


(1") 

a,.S 
-^=2^.  +  2.. 

(2") 

^^-2«  +  23' 

(3") 

s 

2"-' 

,  ([^-1]") 

(^") 

B) 


Now,  since  they  all  together  had  as  much  money  when 
they  left  off  playing,  as  they  had  before  playing,  it  follows, 
that 

5=ai  +  a2  +  «3+ 04  +    -     -     -     -     fln-     (C) 

If  this  value  of  S  be  substituted  in  the  system  of  equations 
(B),  we  shall  then  have  the  values  of 

Ii,    To?     3-3)      -  -  -  -  -      Xnj 

in  terms  of  known  quantities. 
If  we  have  the  relation 

fli  =  02  =  03  =  oi  =     -     -     -     -     =  flu, 


jr.   t>^-*Y\ 


^ 


SIMPLE   EQUATIONS  101 

then  (C)  will  give 

■ind  the  system  of  equations  (B)  will  then  become 


\2   '   2"/     1 


If,  in  (D),  we  suppose  7i  =  5  and  ai  =  32,  we  shall  have 
Xi  =  8l;  a:2=41;  X3=:21;  X4=ll;  Z5=6. 

The  above  supposition  causes  our  question  to  agree  with 
Ques.  13,  Chap.  XII,  Higher  Arithmetic. 

/'w+x  +  3/  =  13,     (1)) 
\xi  +  x-^z=\l,     (2)( 
7.  Given  ^^_|_^_^^^jg^     ^4^  ?  to  find  w,  x,  y,  and  z, 

Dividing  the  sum  of  these  four  equations  by  3,  we  obtain 
u-{-x-\-y-\-z  =  22,.  (5) 

From  (5),  subtracting  successively,  (4),  (3),  (2),  and  (1), 
and  we  find 


S.   Given  <        ^  ^  '  Mo  find  x  and  y. 

I    x  +  5y=191,i  ^ 


Ans. 


x=  16. 
y=36. 


102  SIMPLE    EQUATIONS. 

9.  Given  <         -  '^  ^    ''>tofindxand 

^        9y—  347=  5a:-420,      i 

)3^23 


,        ,a=56. 

Ans. 


10.  Given  <  ^  +  2/      3a  +  x   I,  ^^  fl^d  a:  and  y. 

\    ax  -\-  2by  =dy    ) 

\  ^  = 5 ^• 

'  ^~  36 

11.  A  and  B  possess  together  a  fortune  of  $570.  If  A's 
fortune  were  3  times,  and  B's5  times  as  great  as  each  really 
is,  then  they  would  have  together  $2350.  How  much  had 
each  ? 

Ans.  A  $250,  B    $320. 

12.  Find  two  numbers  ot  the  following  properties  :  When 
the  one  is  multiplied  by  2,  the  other  by  5,  and  both  products 
added  together,  the  sum  is  =31  ;  on  the  other  hand,  if  the 
first  be  multiplied  by  7,  and  the  second  by  4,  and  both  pro- 
ducts added  together,  we  shall  obtain  68. 

Ans.  The  first  is  8,  and  the  second  is  3. 

13.  A  owes  $1200,  B  $2550  ;  but  neither  has  enough  to 
pay  his  debts.  Lend  me,  said  A  to  B,  j  of  your  fortune, 
and  I  shall  be  enabled  to  pay  my  debts.  B  answered,  I  can 
discharge  my  debts,  if  you  will  lend  me  ^  of  yours.  What 
was  the  fortune  of  each  ? 

Ans.  A's  fortune  is  $900,  and  that  of  B  $2400. 

14.  There  is  a  fraction,  such,  that  if  1  be  added  to  the 
numerator,  its  value  =J,  and  if  1  be  added  to  the  denomi- 
nator, its  value  =  i.     What  fraction  is  it  ? 

Ans.  j^. 


SIMPLE    EQUATIONS.  103 

15.  The  sum  of  two  numbers  is  =  «,  the  quotient  r.rishiL; 
from  the  division  of  the  second,  by  the  first  is  =  /;.  Find 
these  numbers  ? 

Ans.^-p-j,an,lj-^_^-j. 

16.  A,  Bj  C,  owe  together  $2190,  and  none  of  them  can 
alone  pay  this  sum  ;  but  when  they  unite,  it  can  be  done  in 
the  following  ways  :  first,  by  B's  putting  ?■  of  his  property 
to  all  of  A's  ;  secondly,  by  C's  putting  t}  of  his  property  to 
all  B's;  or,  by  A's  adding  §  of  liis  property  to  that  of  C. 
How  much  did  each  possess  ? 

Ans.  A  $1530  ;  B  $1540  ;  and  C  $1170. 

17.  A  and  B  possess,  together,  only  |  of  the  property  of 
C  ;  B  and  C  have,  together,  6  times  as  much  as  A  ;  were 
B  $680  richer  than  he  actually  is,  then  he  would  have  as 
much  as  A  and  C  together.     How  much  has  each? 

Ans.  A,  has  $200  ;  B,  $360  ;  and  C,  $840. 

18.  Three  masons.  A,  B,  C,  are  to  build  a  wall.  A  and 
B,  jointly,  could  build  this  wall  in  12  days;  B  and  C  could 
accomplish  it  in  20  days  ;  but  C  and  A  would  do  it  in  15 
days.  What  time  would  each  take  to  do  it  alone  in  ?  And 
in  what  time  will  they  finish  it,  if  all  three  work  together? 

^^g   \  A  requires  20  days,  B  30,  and  C  60  ; 
^^'  I     all  three  together  require  10  days. 

19.  Three  laborers  are  employed  in  a  certain  work.  A  and 
B  would,  together,  complete  this  work  in  a  days  ;  B  and  C  re- 
quire b  days  ;  but  C  and  A,  only  c  days.  What  time  would 
each  require,  singly,  to  accomplish  it  in  1  And  in  what  time 
would  they  finish  it,  if  they  all  three  worked  together  ? 

Answer, 

.  .  2abc         ,  T.  2abc  , 

A  requires  -r—- davs,    B, • .    days, 

ab  -\-  be  —  ca     '  be  +  ca  —  cb 

^  2abc  ,  T  •     ,  2fl6c          , 

days,:    Jointly,  ^-—^-p— days. 


ca -^  ab  —  be      ''  '  ^  ab  -\-bc  +ca 


104  SIMPLE    EQUATIONS. 

20.  A  certain  number  consists  of  three  digits,  which  are 
in  an  arithmetical  progression.  If  this  number  be  divided 
by  the  sum  of  its  digits,  (that  is,  without  considering  the  value 
they  have  as  tens  and  hundreds,)  the  quotient  is  48  ;  but 
if  198  be  subtracted  from  it,  then  we  obtain  for  the  remain- 
der a  number  consisting  of  the  same  digits  as  the  one  sought, 
but  in  an  inverted  order.     What  number  is  this  1 

Ans.  432. 

21.  A  cistern  containing  210  buckets,  may  be  filled  by  2 
pipes.  By  an  experiment,  in  which  the  first  was  open  4, 
and  the  second  5  hqurs,  90  buckets  of  water  were  obtained. 
By  another  experiment,  when  the  first  w-as  open  7,  and  the 
other  3^  hours,  126  buckets  were  obtained.  How  many 
buckets  does  each  pipe  discharge  in  an  hour.  And  in  what 
time  will  the  cistern  be  filled,  when  the  water  flows  from  both 
pipes  at  once  1 

C  The  first  pipe  discharges  15,  and  the 
Ans.  <     second,  G  buckets  ;  it  will  require  10 
(    hours  for  them  to  fill  the  cistern. 

22.  According  to  Vitruvius,  Hiero's  crown  weighed  20 
lbs.,  and  lost  1^  lbs.,  nearly,  in  water.  Let  it  be  assu- 
med that  it  consisted  of  gold  and  silver  only,  and  that 
20  lbs.  of  gold  lose  1  lb.  in  water,  and  10  lbs.  of  silver, 
in  like  manner,  lose  1  lb.  How  much  gold,  and  how 
much  silver  did  this  crown  contain. 

Ans.   15  lbs.  of  gold,  and  5  pounds  of  silver. 

23.  A  person  has  two  large  pieces  of  iron  whose  weight 
is  required.  It  is  known  that  |  of  the  first  piece  weighs  96 
lbs.  less  than  ^  of  the  other  piece  ;  and  that  |-  of  the  other 
piece  weighs  exactly  as  much  as  ^  of  the  first.  How  much 
did  each  of  these  pieces  weigh  ? 

Ans.  The  first  weighs  720  lbs.,  the  second  512  lbs. 

24.  Two  persons,  A  and  B,  can  together  perform  a  piece 


SIMPLE    EQUATIONS.  105 

of  work  in  16  days.  After  having  laboured  jointly  4  days, 
A  leaves,  and  B  by  laboring  36  days  more,  completes  it. 
How  many  days  Avould  each  separately  require  1 

.         ^  A  requires  24  days, 
I  B  requires  48  days. 

25 .  A  merchant  has  two  kinds  of  wine  ;  if  he  mix  a  gal- 
lons of  the  worst  wine  with  b  of  the  best,  the  mixture  is 
worth  c  dollars  per  gallon  ;  but  if  he  mix  f  gallons  of  the 
worst  with  g  gallons  of  the  best,  then  the  mixture  is  worth 
h  dollars  per  gallon.  What  is  the  price  of  each  kind  of 
wine  per  gallon  '? 

, Price  of  the  worst,  ^     '      ^    " ; \~ ^  '      , 

^       J  ag  —  kf 

"'■     'Price  of  the  best,  ^^ +  ^) 'f-^f^ ^)  ^\ 

26.  Seveial.  detachments  of  artillery  divided  a  certain 
number  of  cannon  balls.  The  first  took  72  and  ^  of  the 
remainder  ;  the  next  144  and  i  of  the  remainder  ;  the 
third  216  and  ^  of  the  remainder  ;  and  the  fourth  288  and 
I  of  the  remainder,  and  so  on  j  when  it  was  found  that  the 
balls  had  been  equally  divided.  What  was  the  number  of 
detachments  and  the  number  of  balls  1 

Ans.  8  detachments,  and  4608  balls 

27.  A  person  has  three  horses  and  a  saddle,  which  of 
itself  is  worth  220  dollars.  If  he  put  the  saddle  on  the 
back  of  the  first  horse,  it  will  make  his  value  equal  to  that 
of  the  second  and  third  ;  but  if  he  put  it  on  the  back  of 
the  second  horse,  it  will  make  his  value  double  that  of  the 
first  and  third  ;  and  if  he  put  it  on  the  back  of  the  third 
horse,  it  will  make  his  value  triple  that  of  the  first  and 
second.     What  is  the  value  of  each  horse  ? 

Ans.  20,  100,  and  140  dollars 
13 


106  SIMPLE   EQUATIONS. 

ELIMINATION  BY  INDETERMINATE  MULTIPLIERS. 

(86.)    Suppose  we  wish  to  find  x  arid  y  from  the  equa- 
tions 

2a: +  33/ =13,  (1) 

5x  +  43/  =  22.  (2) 

Multiplying  (1)  by  ??i,  we  find 

2mx  +  Zmy  =  13m.  (3) 

Adding  (2)  and  (3)^  we  have 

(2wi+5)xH-(37?i+4)y  =  13/n  +  22.      (4) 

Assume  3m  -[-4^0  which  gives 

ni  =  -h  (5) 

This  value  of  m  causes  equation  (4)  to  become 

13m -f  22       -  ,„. 

^=-^ r-^=2.  (6) 

2m +  5 

Again,  if  we  had  assumed  2m  +  5  =  0,  which  would  have 

given 

m  =  -f,  (7) 

then  equation  (4)  would  have  become 

,  =  13m±12_3.  (8) 

^         3m  +  4  ^  ^ 

Now,  returning  to  our  former  equations,  we  will  subtract 
(2)  from  (3)  ;  we  thus  obtain 

(2m  —  5)  :r  +  (3/n  —  4)  y  =  13m  —  22.       (9) 
Assume  3m  —  4=0,  which  gives 

m  =  f. 
This  value  of  m  causes  (9)  to  become 
13m  —  22       ^ 


(10) 
(11) 


2m  — 5 

Again,  assume  2m  —  5  =  0,  which  gives 

m=^  (13) 

This  causes  (9)  to  become 


SIMPLE    EQUATIONS.  107 

13m  +  22^  (13) 

These  values  of  a:  and  y  are  the  same  as  just  found. 

It  is  evident  that  had  we  multiplied  (2)  by  m,  and  then 
added,  or  subtracted  the  result  from  (1),  we  should  then 
have  found,  in  a  similar  manner,  the  same  values  for  x 
and  y. 

(87.)  We  will  now  apply  this  method  to  the  two  literal 
equations, 

Z:x+Fi2/  =  A,  "         (1) 

Zoa;  +  Y.y  =  M^.  (2) 

In  these  equations  the  capital  letters  are  supposed  to  be 
known,  and  their  subscript  numerals  indicate  the  equation 
to  which  they  belong.     Thus, 

Xo  is  the  coefficient  of  a:  in  the  second  equation. 

Yi  is  the  coefficient  of  y  in  the  first  equation. 

^2  is  the  absolute  term,  or  the  term  independent  of  x  and 
y  in  the  second  equation. 

Returning  to  our  equations,  we  will  multiply  (1)  by  m 
and  add  the  result  to  (2) ;  we  thus  obtain 

(Zim  +  Zo)  X  +  (  Yijn  +  F,)  y  =  .^^m  +  ^s-      (3 ) 
Assume  Yim  +  F)  =  0,  which  gives 

m  =  — -.  (4) 

This  causes  (3)  to  become 

( Xim  -\-Xo)x  =  ^im-{-  ^2,  (5 ) 

which  gives  immediately 

^ ~  Xim  +  X2~'XjY2  —  A'o Yi  ^  ^ 

Assume  Aim  -f-  Xo  =:  0,  which  gives 


108  SIMPLE   EQUATIONS. 

»  =  -|  (7) 

This  value  of  m  causes  (3)  to  become 

_  Aim  +  >^2  _  Ji-iXj  —  AyX-2 

Hence,  the  values  of  x  tind  y  are 


(S) 


A2X1 — AiX>2 


(9) 


These  values  of  x  and  y  may  be  considered  as  comprising 
the  solution  of  all  simple  equations  combining  only  two  un- 
known quantities.  If  we  wish  to  adapt  this  general  solution 
to  the  equations 

2x  +  3y  =  13, 
5x  +  4y  =  22  ; 

we  must  call 

^1=13;  ^0  =  22. 

Fi=:3;    72  =  4. 

These  values  substituted  in  (9),  give 
x  =  2',  y  =  3. 

(88.)  As  a  still  farther  illustration  of  the  method  of  elimi- 
nation by  means  of  indeterminate  multipliers,  we  will  pro- 
ceed to  the  solution  of  three  simultaneous  simple  equations, 
involving  three  unknown  quantities  x,  y  and  z;  and  we  will 
continue  to  make  use  of  the  notation  by  the  assistance  of 
subscript  numbers. 

Let  the  equations  be  as  follows  : 


SIMPLE    EQUATIONS,  109 

A>-}-  \\y^Z^z  =  A^,  (1) 

Xcx  +  Y.,y  +  Zo2  =  ^o,  (2) 

^3x4-  F3y  +  Z,z  =  ^3.  (3) 

In  these  equations,  as  in  those  of  the  last  example,  the 
capital  letters  X,  F,  Z,  are  the  coefficients  of  their  corres- 
ponding small  letters.  The  small  numerals  placed  at  the 
base  of  these  coefficients  correspond  to  the  particular  equa- 
tion to  which  they  belong.  Thus  X-i  is  the  coefficient  of  x 
in  the  second  equation  ;  F3  is  the  coefficient  of  y  in  the 
third  equation  ;  Zi  is  the  coefficient  of  z  in  in  the  first  equa- 
tion, and  so  for  the  other  coefficients.  The  letter  A  is  used 
to  denote  the  right-hand  members  of  the  equations,  or  the 
absolute  terms  ;  the  subscript  numbers  in  this  case  also  de 
note  the  equation  to  which  they  belong. 

This  kind  of  notation,  by  use  of  subscript  numbers,  is  very 
natural  and  simple,  and  combines  many  advantages  over  the 
ordinary  methods. 

Having  explained  this  method  of  notation,  we  will  now 
proceed  to  the  solution  of  our  equations. 

If  we  multiply  (1)  by  ?n,  and  (2)  by  7j,  and  then  add  the 
results,  we  shall  obtain 


(Xiwi -f  X-ra) X  -f  ( Ym  +  Fon) y 
-j-  {Ziin  -\-  Z'in)z  =  Jlym  -f-  A^n. 


(4) 


From  (4)  subtracting  (3),  we  find 

(  Xim  -\-  X<in  —  Xz)x+{  Yitn  -\-  Y^n  —  F3)  y 
-\-[Zim-\-Zon  —  Z3)s  =  A\m-{-Aon — A^. 

In  order  to  cause  y  and  z  to  vanish  from  this  equation,  we 
will  assume 


(5) 


Y,m  +  Ym=Y,,  (6) 

Z^w  -i-Z<<n  —  Zz.  (7) 


110  SIMPLE    EQUATIONS. 

This  assumption  causes  (5)  to  become 

{Xym  +  X^:>a  —  X2)x  =  Aini  +  Ji2n  —  .U.  (8) 

Therefore, 

.    Ai'in  -f-  A-iU  —  Ai  /  Qv 

Xxm  -j-  X<in  —  X^ 

We  must  now  find  the  values  of  m  and  w,  by  aid  of  con- 
ditions (6)  and  (7)  ;  for  this  purpose  we  will  compare  them 
with  (1)  and  (2),  (Art.  87).  Now,  !n  order  to  make  (6) 
and  (7)  agree  with  (1)  and  (2)  respectively,  we  must  change 
X  to  m,  y  to  n;  X,  to  F, ,  Xg  to  Z,,  F,  to  F,,  Y^  to  Zj, 
A ,  to  F3,  A^  to  Z3.  Making  these  same  changes  in  equa- 
tions (9)  of  Art.  87,  we  obtain 

F3Z2  —  Z3  Y-2 


(10) 
(11) 

(12) 

a3) 


Substituting  these  values  of  m  and  71,  in  (9),  we  readily 
find 


F1Z2  —  Z1F2 

^ZaFi— F3Z1 
""       FiZo— Z1F2' 

Arranging  the  terms  alphabetically,  we  have 
F3Z2— F2Z3 


F1Z2- 

-F2Z1 

F1Z3- 
f;z2-^ 

-F3Z1 

-Y,Z, 

SIMPLE    EQUATIONS. 


Ill 


:i;i^ 


1 

1 

1 

1 

^ 

tn 

^ 

1 

g 

>^"  i 

I  I 

N  o 

^"  I 

+  : 

«  o 

>^'  I 

^  i 

4-  ^ 

CO  "^ 

tS3  - 


+  + 


N 


I      I 


I 

I      I 


+  +       +  + 


+  + 


119  SIMPLE   EQUATIONS. 

(89).  We  will  now  proceed  to  point  out  some  remark- 
able relations  in  the  combinations  of  the  letters  marked 
with  subscript  numbers,  as  given  by  equations  (15),  (16), 
and  (17). 

I.  The  denominator,  which  is  common  to  the  three 
expressions,  is  composed  of  six  distinct  products,  each 
consisting  of  three  independent  factors.  Three  of  these 
products  are  positive,  and  three  are  npgative. 

II.  The  letters  forming  the  different  products  of  this  com- 
mon denominator  being  always  arranged  in  alphabetical 
order,  X,  F,  Z,  we  remark  that  the  subscript  numbers  of 
the  first  product  are  1,  2,  3.  Now,  if  we  add  a  unit  to 
each  of  these  numbers,  observing  that  when  the  sum 
becomes  4  to  substitute  1,  we  shall  obtain  2,  3,  1,  which 
are  the  subscript  numbers  of  the  second  product.  Again, 
increasing  each  of  these  by  1,  observing  as  before,  to  write 
1  when  the  sum  becomes  4,  we  find  3,  1,2,  which  are  the 
subscript  numbers  of  the  third  product.  If  we  increase 
each  of  these  last  numbers  by  1,  observing  the  same  law, 
we  shall  obtain  1,  2,  3,  which  are  the  subscript  numbers 
belonging  to  the  first  product.  A  similar  method  of  chang- 
ing has  already  been  noticed  under  Art.  85. 

What  we  have  said  in  regard  to  the  subscript  numbers  of 
the  positive  products,  applies  equally  well  in  respect  to  the 
negative  products. 

III.  The  numerator  of  the  expression  for  x,  may  be 
derived  from  the  common  denominator  by  simply  substi- 
tuting J?  for  Xj  observing  to  retain  the  same  subscript 
numbers. 

The  numerator  of  the  expression  for  y  may  be  derived 
from  the  common   denominator  by  substituting  A  for  F, 
observing  to  retain  the  same  subscript  numbers. 
14 


SIMPLE    EQUATIONS.  113 

In  the  same  way  may  the  numerator  of  the  expression 
for  z  be  found  by  changing  Z  of  the  denominator  into  A, 
retaining  the  same  subscript  numbers. 

(90.)  We  will  now  proceed  to  show  how  these  expres- 
sions, for  X,  y,  and  c,  can  be  obtained  by  a  very  simple  and 
novel  process,  which  is  easily  retained  in  the  memory,  and 
which  is  applicable  to  all  simple  equations  involving  only 
three  unknown  quantities. 

Writing  the  coefficients  and  the  absolute  terms  in  the 
same  order  as  they  are  now  placed  in  equations  (1),  (2), 
(3),  we  have 


^l 

Fi 

Zi   =:    A, 

x> 

Fa 

Zo  =  A. 

X; 

Fa 

Za  =  A; 

i 


w 


Now,  all  the  products  of  the  common  denominator  can 
be  found  by  multiplying  together  by  threes,  the  coefficients 
which  are  found  by  passing  obliquely  from  the  left  to  the 
right,  observing  that  if  the  products  obtained  by  passing 
obliquely  downwards,  are  taken  positively,  then  those  form- 
ed by  passing  obliquely  upwards  must  be  taken  negatively, 
and  conversely.  This  is  in  accordance  with  the  property 
of  the  negative  sign.  In  the  present  case  the  products 
formed  by  passing  obliquely  downwards,  are  taken  posi- 
tive. 

In  this  sort  of  checker-board  movement,  we  must  ob- 
serve that  when  \ve  run  out  at  the  bottom  of  any  column, 
we  must  pass  to  the  top  of  the  same  column ;  and  when 
we  run  out  at  the  top,  we  must  pass  to  the  bottom  of  the 
same  column. 

This  method  is  most  readily  performed  upon  the  black- 
board, by  drawing  oblique  lines  connecting  the  successive 
factors  of  the  different  products. 


114  SIMPLE    EQUATIONS. 

We  will  trace  out  this  sort  of  oblique  movement. 

Commencing  with  .Xi,  we  pass  obliquely  downwards  to 
Foj  and  thence  to  Z3,  and  thus  obtain  the  positive  product 
of  Xr  Yi  Z3. 

Commencing  with  X^,  we  pass  obliquely  downwards  to 
F3,  and  since  w'e  have  now  run  out  with  the  column  of  Z's 
at  the  bottom,  w^e  pass  to  Z],  at  the  top  of  the  column,  and 
thus  obtain  the  positive  product  X2F3Z1. 

Again,  commencing  with  X3,  we  pass  to  Fi,  and  thence 
obliquely  downwards  to  Zo,  and  find  the  positive  product 
XzYiZ'Z. 

Now,  for  the  negative  products  we  make  similar  move- 
ments obliquely  upwards. 

Thus,  commencing  with  Xi,  we  pass  to  Y3,  and  thence 
obliquely  upwards  to  Zo,  and  find  the  negative  product 
X1F3Z,. 

Commencing  with  A'o,  we  pass  obliquely  upwards  to  Fi, 
and  thence  to  Z3,  and  find  the  negative  product  X^  Fi  Z3. 

Again,  commencing  with  A3,  we  pass  obliquely  upwards 
to  Fi,  and  thence  to  Zi,  and  thus  obtain  the  negative  pro- 
duct A3F2Z1. 

Having  thus  obtained  the  denominator  which  is  common 
to  the  values  of  x,  y,  z;  we  may  find  the  numerator  of 
the  value  of  a-,  by  supposing  the  c/Ts  to  take  the  place  of 
the  A's,  and  then  to  repeat  our  checker-board  movement. 
By  changing  the  F's  into  the  ^^'s,  we  shall  find  the  numera- 
tor of  the  value  of  y  ;  and  by  changing  the  Z's  into  jJ's 
we  shall  find  the  numerator  of  the  value  of  z. 

(91.)  We  will  now  illustrate  this  method  of  solving 
simple  equations  containing  only  three  unknown  quantities, 
by  a  few  examples. 


SIMPLE    EQUATIONS.  \]i) 

2x  -j-  3y  -{.  4z  =  16,  -\ 
Given    ^3x  -{-  5y  -\~  Iz  =  26,  >  to  find  x,  y,  and  z. 
4x  -f  23/  +  3z  =  19,  ) 

We  will  first  find  the  common  denominator. 

Positive  Products.  Negative  Products. 

2X5X3=30  2X2X7  =  — 28 

3x2x4  =  24  3x3x3  =  — 27 

4X3X7  =  84  4X5X4=  — 80 


138  —  135 

—  135 


3  =  common  denominator. 

We  have  for  the  numerator  of  x  the  following  operation 

Positive  Products,  Negative  Products. 

16X5X3  =  240  16X2X7  =  — 224 

26X2X4  =  208  26x3x3  =  — 234 

19x3x7  =  399  19X5X4=— 380 


847  —  838 

838 


9  =  numerator,  for  x. 

To  find  the  numerator  for  y,  we  have 

Positive  Products.  Negative  Products. 

2X26X3  =  156  2x19x7  =  — 266 

3X19X4=228  3x16x3  =  — 144 

4X16x7=448  4x26X4=  — 416 

832  —  826 

—  826 

6  =  numerator,  for  y. 


116  SIMPLE    EQUATIONS. 

To  find  the  numerator  for  c,  we  have 

Positive  Products.  Negative  Products. 


2x5x19=  190 

2X2X26=— 104 

3X2X16=:    96 

3X3X19  =  — 171 

4X3X26  =  312 

4X5X16  =  — 320 

598 

—  595 

—  595 

3  = 

=  numerator,  for  z. 

Hence,                         x 

—  ti 

3 

=  3. 

y 

=     1 

=  2. 

z 

3 

=  1. 

When  some  of  the  coefficients  are  negative,  we  must  ob- 
serve the  rule  for  the  multiplication  of  signs. 

r  2a;  +  42/  — 3r  =  22,  ^ 
2.  Given  ^  4x  —  2y  +  5r=  IS,  >  to  find  x,  y,  and  z, 
(  6x-{-ly—   z  =  62,  ) 

To  find  the  common  denominator,  we  have 

Positive  Products.  Negative  Products. 

2X  — 2X  — 1=  4  2X  7x  5=  — 70 
4X  7X  — 3=-84  4X  4X— 1=  16 
6X      4X      5=    120        6X  — 2X  — 3=— 36 

40  -90 

-90 

—  50  =  common  denominator. 


SIMPLE    EQUATIONS.  117 


Positive  Products.  Negative  Products. 


22X- 

-2X  — 1=    44 

22  X 

7X   5=  — 770 

18  X 

7X  — 3  =  — 378 

18  X 

4X  — 1==   72 

63  X 

4X   5=  1260 

63X- 

-2X  — 3^— 378 

926 

-1076 

—  1076 

—  150  =  numerator,  for  x. 


—   50 

Proceeding  in  a  similar  way,  we  find  the  values  of  y  and  z. 


to  find  X,  y,  and  z. 
'J  ) 

We  will  arrange  the  coefficients,  omitting  the  unknown 
quantities,  observing  also  to  write  0  for  such  terms  as  are 
wanting. 

This  arrangement  being  made,  we  have 
1     ^     0     =     a 
0     1     1     =     a 
i     0     1     =     a 

Positive  Products.  Negative  Products. 

1X1X1=  1  lXOXi  =  0 

0X0X0=0  OxiXl  =  0 

iXiXi  =  ^V  1X1X0  =  0 

2  A  =  common  denominator.       0 

For  the  numerator  of  x,  we  have 


118  simple  equations. 

Positive  Products.  Negative  Products. 

0X1X1=  a  axOXi=    0 

aXOxO=  0  aX\Xl  =  —  \a 

aX}sX\  =  la  aXlXO=    0 


jfl  =  numerator,  for  x. 

Hence,  x=%a-^%\=^  \la. 

By  a  similar  process  is  the  value  of  y  and  z  fouml. 

C  x-\-a{y-\-z)=m^  ^ 
4.  Given  }^  y  -\-b{x  -\-  z)=n,   >  to  find  rr,  y,  and  2. 

These  coefficients,  being  properly  arranged  give. 


1     a 

a 

1=     m 

b     1 

b 

=     n 

c     c 

1 

=     P 

siTivE  Products. 

Negative  Products 

1X1X1=     1 

\XcXb=—hc 

b  XcXa=abc 

ftXaXl=— aft 

c  XaXb=abc 

cXlXa=— ac 

\-\-2abc  — ab  —  ac — be 

■ab  —  ac  —  be 


1  -\-2abc  —  ab  —  ac  —  bc  =  common  denominator. 
For  the  numerator  of  .t,  we  have 


simple  equations.  119 

Positive  Products.  Negative  Products. 

mxlxl=    m  mxcxh:=  —  bcia 

n  xc  xa  =  (ten  n  xax\=-  —    an 

p  xaxb  =: ahp  ^  x  1  X  o  =:  —    nj; 


m  -j-  acn  -\-  ahp  —  hem  —  an  —  ap 

hem  —  an  —  ap 


m-\-acn-{-ahp  —  hem  —  an  —  ap  =  numeralor  of  x. 

Tj  m-\-acn-\-abp  —  hem  —  an — ap 

1  -|-  2ahc  —  ah  —  ae  —  he 

If  to  this  expression  for  x  we  apply  the  principle  of  per- 
mutation, as  already  explained,  by  advancing  the  letters  one 
place  lower  in  the  alphabetical  scale,  we  shall  find 

n  -f-  hap  -j-  hem  —  can  —  hp  —  hm 

l'-\-2hca  —  he  —  6c  —  ea 
Again,  permuting  this  expression,  we  have 

p-\-  ehm  -j-  ean  —  ahp  —  cm  —  C7i 

l-\-2eab  —  ca  —  ch  —  ah 

This  solution  is  far  shorter  than  the  one  given  on  page 
94,  and  the  expressions  for  x,  y,  and  z,  are  far  more  sim- 
ple. 

We  may  remark,  that  the  denominators  of  the  above  ex- 
pressions are  common,  as  they  must  of  necessity  be,  in  vir- 
tue of  the  general  results  given  by  Equations  (15),  (16),  (17), 
on  page  111. 

5.  A,  B,  and  C,  owe  together  (a)  $2190,  and  none  of 
them  can  alone  pay  this  sum  ;  but  when  they  unite,  it  can 
be  done  in  the  following  ways  :  first,  by  ]3's  putting  5  of 
his  property  to  all  of  A's  ;  secondly,  by  Cs  putting  j  of 
his  property  to  all  of  B's  ;  or  by  A's  putting  I  of  his 
property  to  all  of  Cs.     How  much  was  each  worth  1 


120  SIMPLE    EQUATIONS. 

Let  X,  y,  and  c,  represent  -what  A,  B,  and  C,  were  re- 
spectively worth. 

Then  we  shall  have  these  conditions, 
X  +  ^y  =  a, 
y  +  f  c  =  a, 
z  +  |x  =  a. 

Clearing  these  of  fractions,  and  arranging  the  coefficients, 
we  have 

7     3     0     =     7a 

0     9     5     =     9a 

2     0     3     =     3a 

Positive  Products.  Negative  Products. 

7  X  9X  3=1=189  7  X  0  X  5=0 

0  X  0  X  0=      0  0  X  3  X  3=0 

2x3x5=   30  2X9X0  =  0 

219  =  common  denominator.     0 


Positive  Products.  Negative  Products. 

7a  X  9  X  3  =  189a  7a  X  0  X  5  =        0 

9aX0x0=      0  9ax3x3  =— 8la 

3a  X  3  X  5  =    45a  3a  X  9  X  9=      0 

234a  —81a 

—81a 

153a  ::=  numerator  of  x- 


153a  _  153X2190 
"219  ~        219 

For  the  numerator  of  y,  we  find 


Hence,  x  = -^  =        ^^^        =  1530. 


simple  equations.  121 

Positive  Products.  Negative  Products 

7  X  9a  X  3  =  189a  '         7  X  3a  X  5  =  —  105a 
0x3aX0==      0  0x7ax3=  0 

2  X  7a  X  5  =    70a  2  X  9a  X  0  =  0 


259a  —  105a 

105a 


154a  =  numerator  of  y. 


154a 
Hence,  y  =  ^^  =  1540. 


For  the  numerator  of  z,  we  have 

Positive  Products.  Negative  Products. 


7x9x3ai=189a                      7x0x9a  = 

0 

0x0x7a=      0                         0x3x3a  = 

0 

2x3X9a=    54a                       2x9x7a  = 

—  126a 

243a 

—  126a 

—  126a 

117a  =  numerator  of  z. 

Hence,  z  =  ^19- =  11^0- 

Collecting  the  results,  we  find  that 

A  was  worth  $1530, 

B     "        "     $1540, 

C     "        "      $1170.      . 

The  student  will  find,  after  a  little  practice  in  this  method 
that  it  is  much  more  simple  than  would  at  first  sight  seem 

Whenever  some  of  the  coefficients  are  zeros,  as  in  th« 
3d  and  5th  examples,  the  work  is  much  abridged,  as  in  this 
caee  some  of  the  products  must  become  zero 
15 


INVOLUTION. 


CHAPTER  IV. 

INVOLUTION,  EVOLUTION,  IRRATIONAL  AND 
IMAGINARY  QUANTITIES. 


INVOLUTION. 

(92.)  The  process  of  raising  a  quantity  to  any  proposed 
power  is  called  Involution. 

When  the  quantity  to  be  involved  is  a  single  letter,  it  is 
involved  by  placing  the  number  denoting  the  power  above 
it  a  little  to  the  right.     (Art.  11.) 

After  the  same  manner  we  may  represent  the  power  of 
any  quantity,  by  enclosing  it  within  a  parenthesis,  and  then 
treating  it  as  a  single  letter. 
Thus, 

the  second  power  of  mx  =^  {mxy, 
the  third  power  of  a  +  6  =  (a  -f-  by, 
the  fourth  power  of  3m  -{-  y=  (3m  +  y)*, 
&c.,  &c. 

CASE   I. 

(93.)   To  involve  a  monomial,  we  obviously  have  this 


INVOLUTION.  123 


RULE. 

I.  Raise  the  coefficient  to  the  required  power^  by  actual 
multiplication. 

II.  Raise  the  different  letters  to  the  required  power  by 
multiplying  the  exponents^  which  they  already  have,  by  the 
number  denoting  the  power,  observing  that  if  no  exponent 
is  written ,  then  one  is  always  understood.  To  this  power 
prefix  the  power  of  the  coefficient. 

Note. — If  the  quantity  to  be  involved  is  negative,  the 
signs  of  the  even  powers  must  be  positive,  and  the  signs  of 
the  odd  powers  negative.     (Art.  29.) 


1.  What  is  the  square  of  Zax^l 
Here  the  square  of  3  equals 

3==  3x3  =  9. 
Considering  the  exponent  of  a,  in  the  expression  ax^,  i 
one,  ve  find  a'x^  for  the  square  of  ax^. 
Therefore  we  have 

{'iax^Y=2aHK 

2.  What  is  the  fifth  power  of  —  2ab^  ? 

Ans.   {—2ab^)^=  —  ?,2a^b'' 

3.  What  is  the  fourth  power  of  — -  xy~^? 

o 

Ans.  (_lx2/-')^==^^:r*3/-«, 
which  by  Art.  49,  is  the  same  as 


124  INVOLUTION.    , 

4.  What  is  the  seventh  power  of  — ar^xl 


Ans. -a-V  =  — ^. 

5. 

What  is  the  third  power  of  x^jr'  ? 

Ans.  x^jr'  =  -^ 

6. 

What  is  the  nth  power  of  — 2x-3  y^  1 

Ans.   ±2"a:-3Y''  =  i:-^ 

7. 

What  is  the  square  of  —  7a;  -  *  y  -  ^  ? 

Ans.  49x-^y-=i?, 

8.  What  is  the  third  power  of  —  -  x'^y~^  ? 

1  x^ 

Ans. r  x^y~^^  = 


125     "  125y'^ 

9.  What  is  the  seventh  power  of  —  m^xz-^  ? 

7 

Ans.  — m^x''z~'^. 

10.  What  is  the  fourth  power  of  —  -  n-^y^  ? 

Ans.  Q^n^Y^- 


CASE   II. 


(94)  When  the  quantity  is  compound,  we  can  write  the 
different  powers  by  the  aid  of  rules  which  we  will  hereafter 
point  out.     (See  Binomial  Theorem.) 

At  present  we  will  content  ourselves,  by  involving  com- 
pound expressions  by  actual  multiplication,  according  to 
Rule  under  Art.  33. 


INVOLUTION.  125 


EXAMPLES. 

1.   Find  the  seconil  power  of  x  -{•  V  — 

x-{-y  —  z 

X'  -|-  XT/ XZ 

—  XZ  —yz-{- 


Ans.  =  x'^-\-  2xy  —  2xc+  y^  —  2yz  -f-  z'-. 

2.  Find  the  fifth  power  of  a  -f-  ^j  as  well  as  all  the  lower 
powers  of  the  same. 
{a-\-by  =  a-^b. 
a-\~b 


a-  +  ab 
-^ab-\-b'' 

(a-\-b)'  =  a^-}'2ob  -\-  b^. 
a  +6 


a3  -f  2a'b  -f-  ab" 

a"b  -i-  2ab^  4-  63 

{a^by  =  a^  +  Sa-b  -\-  3ab-  +  b\ 
a    -{-b 


a^  +  3a'b  +  3a'h^  +  ab' 

a^b  +  3a=6=  +  3a6»  +  ^* 

(a  4-6)^  =  a^  4-  4a86  +  60=6^  +  4a6^  +  ft-*, 
a   4-6 

a^J  _|_    4a^6-  +    Ga^ft*  +  4fl6*  -f •  6* 


126  INVOLUTION. 

3.  Find  the  fifth  and  lower  powers  of  a  — b, 
{a  —  b)'=a  —  b. 
a —  b 


-br= 

-b)*= 

a^- 

-ab 
-ab  +  b- 

-b' 

{a- 

a  — 

■'2ab-j-bK 
-b 

a«- 

-   a'b-\-2ab^- 

{a- 

a  - 

-  2a%  -f-  3a6*— 
-b 

b\ 

a*- 

-3a'b-\-3a^'^- 
-  a'b  +  3a-b^- 

-ab 
Sal 

3 

{a- 

a  — 

-4a'b-{-6a^b''- 
-b 

-4a 

b'  +  b*. 

«'- 

~ia*b-{-    6a^^ 

— 

4:aH'+  ab* 
ea^^-\-  4ab*—  6» 

(a_  J)  «=  a'-~5a*b  +  lOa^b^—  10a^^-\-  5ab*—  b' 

4.  What  is  the  cube  of  c  —  x  ? 

Ans.  o^ — 3a^a: -j- 3aa:^ — x^. 

5.  What  is  the  square  of  m-{-n  —  x'f 

Ans.  m^-\-  2mn  —  2mx  +  n^ —  2nx  -\-  x^. 

6.  What  is  the  fourth  power  of  3x  —  2y? 

Ans.  81x^—216x^1/  +  216x^y^—  96xy«  +  16y*. 

7.  What  is  the  square  of  a-\-b? 

Ans.  a"+2ab-i-.b^. 

8.  What  is  the  square  of  a  +  6  +  c  ? 

Ans.  ai-\-  2ab  +  2ac  -{-b^  +  2bc  +  c^. 


EVOLUTION.  127 


EVOLUTION. 

(95.)  Evolution  is  the  extracting  of  roots,  or  the 
reverse  process  of  involution. 

When  the  quantity  whose  root  is  to  be  found  is  a  single 
letter,  the  operation  is  denoted  by  giving  it  a  fractional  ex- 
ponent, the  denominator  of  which  denotes  the  degree  of 
the  root.     (Art.   14.) 

And  in  the  same  way  we  may  denote  the  extraction  of  a 
root  of  any  quantity  or  expression,  by  enclosing  it  within 
a  parenthesis,  and  then  treating  it  as  a  single  letter. 

Thus,  the  second  root  of  my=  {my)'^, 

the  third  root  of  x  -\-  y  ^  {x  -\-  yY , 

i_ 
the  fourth  rpot  of  2x  —  3?/  =  (2x  —  3y)  * , 

J. 
the  nth  root  of  a  —  ft  =  {a  —  b)" , 
&c.,  &., 

CASE   I. 

(96.)  To  extract  a  root  of  a  monomial,  we  obviously 
have  the  following 

R  U  L  E  . 

I.  Extract  the  required  root  of  the  coefficient^  by  the  i/sual 
arithmetical  rule.      When  the  root  can  not  be  accurately  oh- 


128  EVOLUTION. 

tained,  it  may  be  denoted  by  means  of  a  fractional  exponent, 
the  same  as  in  the  case  of  a  letter. 

II.  Extract  the  required  root  of  the  different  letters,  by 
multiplying  the  exponents  which  they  already  have  by  the 
fractional  exponent  denoting  the  required  root.  To  this 
root  prefix  the  root  of  the  coefficient. 

Note. — Since  the  even  powers  of  all  quantities,  whether 
positive  or  negative,  are  positive';  it  follows  that  an  even 
root  of  a  negative  quantity  is  impossible,  and  an  even  root 
of  a  positive  quantity  is  either  positive  or  negative. 

We  also  infer  that  an  odd  root  of  any  quantity  has  the 
same  sign  as  the  quantity  itself. 

EXAMPLES. 

1.  What  is  the  square  root  of  64a^6'*x^  ? 

In  this  example,  the  square  root  of  the  coefficient,  64,13 
±8,  where  we  have  used  both  signs. 
And, 

(0^6^)^=  a6V, 

.-.  (64a«6V)^==h8a6V. 

2.  What  is  the  cube  root  of  64:a^x^  ? 

Ans.  Aax'. 

3.  What  is  the  fifth  root  of  —  32x/  ? 

Ans.  — 2x^y^. 

4.  What  is  the  seventh  root  of  —  ax~^  1 

Ans.  — a'^x~-^=z  —  — 

a 
■       X^ 

6.  What  is  the  square  root  of  — 4a^5*? 

Ans.  Impossible. 


V  EVOLUTION.  129 

6.  What  is  the  cube  root  of  21a^b^''  1 

Ans.  3«6'. 

7.  What  is  the  fourth  root  of  16a  -^x-^  ? 

3-    L        L  26' 

Ans.  =b2a-*6*x-'=i-^— 7. 

(97.)  By  comparing  the  operations  of  this  rule,  with  those 
of  rule  under  Art.  93,  we  see  that  involution  and  evolution 
of  monomials  may  both  be  performed  by  one  general  rule, 
of  multiplying  the  exponents  of  the  respective  letters  by 
the  exponent  denoting  the  power  or  root.  We  will  there- 
fore give  the  following  promiscuous  examples,  which  will 
require  the  aid  of  one  or  both  of  these  rules. 

EXAMPLES. 

1.  What  is  the  cube  root  of  the  second  power  of  8a'6^  1 
If  we  first  raise  8a ^6^  to  the  second  power,  it  will  become 
(8o369)2=  64a«&'^ 
extracting  the  third  root,  we  find 

(64a«6»«^^=4a=i«, 
for  the  result  required. 

Again,  first  extracting  the  cube  root  of  Sa^i^,  it  becomes 

raising  this  to  the  second  power,  it  becomes 

the  same  as  before. 

(98.)  Hence,  the  cube  root  of  the  square  of  a  quantity, 
is  the  same  as  the  square  of  the  cube  root  of  the  same  quan- 
tity. 

And  in  general,  the  nth  root  of  the  mth  power  of  a  quan- 
tity^ is  the  same  as  the  mth  power  of  the  nth  root  of  the 
same  quantity. 

17 


130 


EVOLUTION. 


Therefore,  a^  may  be  read,  the  fourth  power  of  the  fifth 
root  o{  a,  or  the  fifth  root  of  the  fourth  power  of  a. 

And  in  the  same  way,  {a  -\-b)'^  is  read,  the  third  power 
of  the  square  root  of  the  sum  of  a  and  6,  or  the  square  root 
of  third  power  of  the  sum  of  a  and  b. 


2.  What  is  the  value  of  (—  3a6 V) 


3  ? 


Ans.  3^ah*^x\ 


3.  What  is  the  value  of  {^a-'^b'xy  ? 

Ans.  ±32a-'b'^x^. 
(99.)    Surd  quantities  may  be  made  to  assume  several 
equivalent  forms  which  require  to  be  read  differently.     As 

3 

an  example,  the  surd  a~*  may   be   written    six    different 
ways,  as  follows  : 

—  I  — )  s 

1.     (M^)     ;  2.     ((«,^)')     ;         3.     ((a-)^)   ; 

4.     ((ah-^;  5.      ^(a-'))';         6.       ((a^)-)'. 

These  six  expressions  are  read  as  follows  : 

1.  The  reciprocal  of  the  fifth  root  of  the  third  powei 
of  a. 

2.  The  reciprocal  of  the  third  power  of  the  fifth  root 
of  a. 

3.  The  third  power  of  the  fifth  root  of  the  reciprocal 
of  a. 

4.  The  third  power  of  the  reciprocal  of  the  fifth  root 
of  a. 

5.  The  fifth  root  of  the  third  power  of  the  reciprocal 
of  a. 


EVOLUTION.  131 

6.  The  fifth  root  of  the  reciprocal  of  the  third  power 
of  a. 

CASE  I. 

(100.)  To  extract  any  root  of  a  polynomial,  we  liave  the 
following  general 

RULE. 

I.  Havhig  arranged  the  polynomial  according  to  the 
powers  of  some  one  of  the  letters^  so  that  the  highest  fower 
shall  stand  first,  extract  the  required  root  of  the  first  term, 
which  will  be  the  first  term  of  the  root  sought. 

II.  Subtract  the  power  of  this  first  term  of  the  root  from 
the  polynomial  .J  and  divide  the  first  term  of  the  remainder.) 
by  the  first  term  of  the  root  involved  to  the  next  inftrior 
power,  multiplied  by  the  number  denoting  the  root;  the  quo- 
tient will  be  the  second  term  of  the  root. 

III.  Subtract  the  power  of  the  terms  already  found  from 
the  polynomial,  and  using  the  same  divisor  proceed  as  before. 

This  rule  obviously  verifies  itself,  since,  whenever  a  new 
term  is  added  to  the  root,  the  whole  is  raised  to  the  given 
power,  and  the  result  is  subtracted  from  the  given  polyno- 
mial :  and  when  we  thus  find  a  power  equal  to  the  given 
polynomial,  it  is  evident  that  the  true  root  has  been  found.. 

1.  What  is  the  fifth  root  of 

a5  -j-  50^6  -f  lOa^i^  _^  \Qa^b^-\-  ba¥  -f  b'  ? 

OPERATION. 

ROOT. 

a»  +  5a^6  +  lOa^ft-  +  lOa^ft  >  4-  dab*  -f-  b'  {a  -\-b 
a* 


5a*)  5a*6 

^a  -\-  by  =  a**  -f  5a*b  -{-  lOa^b^  +  lOa^^  +  5ab*  +  b' 


132  EVOLUTION. 


EXPLANATION. 

We  first  found  the  fifth  root  of  the  first  term  a',  to  be  ff, 
which  we  plated  to  the  right  of  the  polynomial  for  the  first 
term  of  the  root.  Raising  a  to  the  fifth  power  and  subtract- 
ing it  from  the  polynomial,  we  have  Dab  for  the  first  term 
of  the  remainder. 

Since  the  number  denoting  the  root  is  5,  we  raise  the  first 
term  of  the  root,  a,  to  the  fourth  power,  which  thus  becomes 
a',  this  multiplied  by  the  number  derK^ting  the  root,  gives 
5a-*  for  our  divisor. 

Now,  dividing  5a'6  by  5a',  we  get  b,  which  we  write  for 
the  second  terra  of  the  root. 

Involving  this  root  to  the  fifth  power  by  actual  multipli- 
cation, as  was  done  in  Ex.  2,  Art.  94,  we  have 

(a  +  by=  a^+  5a'b  -f  lOa^^^-j-  10a''b^-\-  5ab*+  6*  ; 
which  subtracted  from  the  given  polynomial,  leaves  no  re- 
mainder, so  that  we  know  that  a  -\-  b  \s  the  true  root. 

2.  What  is  the  square  root  of 

4x^  —  16x3  +  24a-  —  16a:  +  4  ? 

OPERATION. 

ROOT. 

4x'  — 16a:3+24j2_l6x+4(2x=^  — 4x-H2 


4:X~) —  ]6x^ 

(2x*— 4xf  =  4x*  —  16x3+i6x=^ 

4x2)8x2 
(Sr*— 4x-f2)=^=4x^  —  16x3+24x2— 16x-f  4 


EVOLUTION.  133 

3.  What  is  the  square  root  of 

16a;*  +  24x='  +  S9x-  -f-  60x  +  100  ? 

Ans.  4x^  +  3x  +  10. 

4.  What  is  the  cube  root  of 

a«  +  3a^  —  3a*  —  lla^  +  6a~  +  12fi  —  8  ? 

Ans.  a-  -(-  a  -s—  2- 

5.  What  is  the  sixth  root  of 

Ans.  a  —  6. 

6.  What  is  the  fourth  root  of 

a*— 4c36  +  6a26-  — 4o53  +  b*  ? 

Ans.  a  —  b. 

(101.)  If  we  carefully  observe  the  law  by  which  a  poly 
nomial  is  raised  to  the  second  power,  we  shall,  by  reversing 
the  process,  be  enabled  to  deduce  a  rule  for  the  extraction  of 
the  square  root  of  a  polynomial,  which  will  be  more  simple 
than  the  above  general  rule,  and  of  more  interest,  since  the 
arithmetical  rule  is  deduced  from  it. 
By  actual  multiplication,  we  find 
(a+6)2  =  a2+2a6+62, 
(a+6+c)-  =  a-+2«&+Z^--f2(«+6)c-4-c% 
{a+b4-c+dy 

=  a2+2a6+62+2(a+6)c  +  c-+2(a+&-fcV-f-aS 
(a-l-6+c+d+e)2 

^  (  a2-h2a6+6'^+2(a+6)c+c2  ) 

I  -\-2{a-\-b  +  c)d+d'-{-2{a+b+c-\-d)e+e'.  i 
&c.  &c. 

From  the  above,  w^e  discover,  that 

(102.)  The  square  of  any  polynomial  is  equal  to  the  square 
of  the  first  term,  plus  twice  the  first  term  into  the  second^ 
plus  the  square  of  the  second ;  plus  tvnce  the  sum  of  the  first 
two  into  the  third,  plus  the  square  of  the  third;  plus  twice 


134  EVOLUTION. 

tke  Stan  ofthejirst  three  into  the  fourth^  plus  the  square  of 
the  fourth;  and  so  on. 

(103.)  Hence,  the  square  root  of  a  polynomial  can  be 
found  by  the  following 

RULE. 

I.  Jlfter  arranging  the  polynomial  according  to  the  powers 
of  some  one  of  the  letters^  take  the  root  of  the  first  term  for 
the  first  term,  of  the  required  root ^  and  subtract  its  square 
from  the  polynomial. 

II.  Bring  down  the  next  two  terms  for  a  dividend.  Di- 
vide its  first  term  hy  twice  the  root  just  found,  and  add  the 
quotient,  both  to  the  root,  and  to  the  divisor.  Multiply  the 
divisor,  thus  increased,  into  the  term  last  placed  in  the  root, 
and  subtract  the  product  from  the  dividend. 

III.  Bring  down  two  or  three  additional  terms,  and  pro- 
ceed as  before. 

EXAMPLES. 

1.  What  is  the  square  root  of 

OPERATION. 

ROOT. 

a-'+2a6H-6-+2(a+6)c+cH-2(flH-i+c)d+d-^[a+6-hc4-d. 


2fl&  +  6--' 
2a6+62 


2(a+6)-|-c      2(a+6)c4-c2 
2(a+6)c+c2 


2(«+6+c)4-d.  2{a-\-b-\-c)dJ^d^ 


EVOLUTION.  135 

2.  What  is  the  square  root  of 

OPERATION. 

RO(T 

4:X^-i-12x^-{-5x*  —  2a:3-|-7a;-  —  2x-\-l  (2x3+30;^  —  x+ 1 . 
4x« 


4x'+3x-    12x^+5x-' 
12x^+9x^ 


4x^H-6x-  — X   —  4X''  — 2x3+7x'^ 
— 4x^  — 6x3-(-  x-^ 

4x3+6x2— 2x+l  4x3-1-6x2— 2x+l 

4x3+6x2— 2x+l 

0 

3.  What  is  the  square  root  of 

X*  —  2xY  —  2a;-  +  2/"  +  2y-  +  1  ? 

Ans.  X-  —  y- —  1. 

4.  What  is  the  square  root  of 

9xy  —  30xhj3  +  25xy  ? 

Ans.  3x-y2 — p,xy. 

5.  What  is  the  square  root  of 

a^  +  2ab  —  2ac  +  i-  —  25c  +  c'  1 

Ans.  a  +  6  —  c. 

6.  What  is  the  square  root  of 

4m-'  — 36m7i  +  81n-'? 

Ans.  2m  —  9/j. 

In  these  examples,  and  in  all  others  where  an  ece?i  root 
is  extracted,  the  terms  of  the  root  may  have  all  their  signs 
changed,  and  still  satisfy  the  questions. 


136  EVOLUTION. 

(104.)  We  will  now  endeavor  to  find  a  particular  rule 
for  the  extraction  of  the  cube  root  of  a  polynomial. 
B-y  actual  multiplication,  we  find 
{a-\-by  =  a^-\-3a-b+3ab^-\-b^, 
{a-\-b-\-cY 

=  as-\-3(rb-\-3ab'-\-b-'-\-'3{a-\-bYc-\-3{a-\-b)c'-\-c', 
{a+b+c+dy 

&c.,  &c. 

(105.)  From  which  we  discover  that 

The  cube  of  any  polynomial  is  equal  to  the  cube  of  the  first 
teryn,  plus  three  times  the  square  of  the  first  into  the  second^ 
plus  three  times  the  first  into  the  square  of  the  second,  plus 
the  cube  of  the  second;  plus  three  times ^  the  square  of  the 
sum  of  the  first  two  into  the  third,  plus  three  times  the  sum 
of  the  first  two  into  the  square  of  the  third^  plus  the  cube  of 
the  third;  plus  three  times  the  square  of  the  sum  of  the  first 
three  into  the  fourth,  plus  three  times  the  sum  of  the  first 
three  into  the  square  of  the  fourth,  plus  the  cube  of  the 
fourth ;  and  so  on. 

(106.)  Now  we  may  reverse  the  above  process,  that  is, 
we  may  extract  the  cube  root  of  a  polynomial  by  the  fol- 
lowing 

RULE. 

I.  Having  arranged  the  terms  of  the  polynomial  accord- 
ing to  the  powers  of  some  one  of  the  letters,  seek  the  cube 
root  of  the  first  term,  which  place  at  the  right  of  the  poly- 
nomial for  the  first  term  of  the  root,  also  place  it  at  the  left 
by  itself,  for  the  first  term   of  a  column,  headed,  first 


EVOLUTION.  137 

COLUMN.  Then  multiply  it  into  itself ,  and  place  the  pro- 
duct for  the  first  term  of  a  column^  headed,  secoi^d  column. 
Again,  multiply  this  last  result,  by  the  same  first  term  of 
the  root  and  subtract  the  product  from  the  first  term  of  the 
polynomial,  and  then  bring  down  the  next  three  terms  of  the 
polynomial,  for  the  first  dividend.  Md  the  first  term  of 
the  root  just  found  to  the  first  term  of  the  first  column,  the 
sum  will  constitute  its  second  term,  which  must  be  multi- 
plied by  the  first  term  of  the  root,  and  the  result  added  to 
the  first  term  of  the  second  column,  for  its  second  term, 
which  we  will  call  the  first  trial  divisor.  The  same  first 
term  of  the  root  must  be  added  to  the  second  term  of  the 
first  column,  forming  its  third  term. 

II.  Divide  the  first  term  of  the  first  dividend  by  the  first 
term  of  the  trial  divisor,  the  quotient  must  be  added  to  the 
root  already  found-,  for  its  second  term,  it  must  also  be 
added  to  the  last  tenn  of  the  first  column,  the  result  will  be 
its  fourth  term, which  must  be  multiplied  by  the  second  term 
of  the  root,  and  the  product  added  to  the  last  term  of  the 
second  column,  which  sum  will  give  its  third  term,  which  in 
turn  must  be  multiplied  by  the  second  term  of  the  rooty  and 
the  product  subtracted  from  the  first  dividend. 

III.  To  the  remainder  bring  down  three  or  four  of  the 
next  terms  of  the  polynomial  for  a  second  dividend.  Pro- 
ceed with  this  second  term  of  the  root,  precisely  as  was  done 
with  the  first  term,  and  so  continue  until  the  entire  polyno- 
mial has  been  exhausted. 


IS 


138 


EVOLUTION. 


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EVOLUTION. 


139 


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140 


EVOLUTION. 


5.  What  is  the  cube  root  of  the  polynomial 

a;6  —  6x^  4-  iDx^  —  20x''  +  15x^  —6a;  +  1 1 

Ans.  x2  — 2a;+l. 

6.  What  is  the  cube  root  of  the  polynomial 

a»  +  12a'x''  —  Sa'x'  —  6a''x  ? 

Ans.  a^  —  2ax. 

7.  What  is  the  cube  root  of 

a'  —  3a'  -j-  ea'  —  la'  -\- 6a^- —  3a  +  1 1 

Ans.  a^  — a-\-l. 

8.  What  is  the  cube  root  of 

x^  +  6x'  +  21x'  +  44x3  +  63ar»  +  54a:  +  27  1 

Ans.  x^-{-2x-\-3. 

(107.)  From  the  above  rule,  for  extracting  the  cube  root 
of  a  polynomial,  we  can  easily  deduce  the  rule  which  we 
have  given  in  the  Higher  Arithmetic  for  the  extraction  of 
the  cube  root  of  a  number. 

This  rule  is  also  particularly  interesting  because  of  its 
close  analogy  to  the  method  of  finding  the  numerical  roots 
of  a  cubic  equation,  as  explained  in  a  subsequent  part  of 
this  work. 


SURD    QUANTITIES,  ^ 


IRRATIONAL  OR  SURD  QUANTI- 
TIES. 

(108.)  An  Irrational  Quantity,  or  Surd,  is  a  quan- 
tity affected  with  a  fractional  exponent  or  radical,  without 
which,  it  can  not  be  accurately  expressed. 

Thus, 

v'B  is  a  surd,  since  the  s-unu  e  root  of  3  can  not  be  accu- 
rately found  5  also  8^  4%  V4,  V5,  &c.,  are  surd  quanti- 
ties. 

REDUCTION    OF    SURDS. 

CASE  I. 

(109.)  To  reduce  a  rational  quantity  to  the  form  of  a 
surd,  we  have  this 

RULE. 

Raise  the  quantity  to  a  power  denoted  by  the  root  of  the 
required  surd;  then  the  corresponding  root  of  this  poicer, 
expressed  by  means  of  a  radical  sign  or  fractional  exponent, 
mil  express  the  quantity  under  the  proposed  form. 

EXAMPLES. 

1.  Reduce  5a  to  the  lorm  of  the  cube  root. 
Raising  5a  to  the  tl.ird  power,  we  have 
(5a)'=r25a^; 
extracting  the  cube  root,  it  becomes 

5a=  Vl25^=(125a^)^ 


142 


SURD    QUANTITIES. 


X 

2.  Reduce  —  to  the  form  of  the  fifth  root. 


3.  Reduce  —  to  the  form  of  the  fourth  root. 


Ans.   —  =1 


4.  Reduce  —  to  the  form  of  the  nih  root 
6* 


oot. 

1 
a-  __  ja-"^ 


CASE    II. 


(110.)  To  reduce  surds  expressing  different  roots  to  equi- 
valent ones  expressing  the  same  root. 


Reduce  the  different  indices  to  common  denominators ;  then 
raise  each  quantity  to  a  power  denoted  by  the  numerator  of 
its  respective  exponent;  afterwards  take  the  root  denoted  hi/ 
the  common  denominator. 

EXAMPLES. 

1.  Reduce  ^/3,  V4,  and  V5  to  surds  expressing  the  same 
root. 

Changing  the  radicals  into  fractional  <jxponents,  they  be- 
come i,  I,  5,  which  reduced  to  a  common  denominator,  are 


SURD   QUANTITIES.  143 

tV?  tV)  fV*  Now,  raising  the  quantities  3,  4,  and  5,  to 
powers  denoted  respectively  by  6,  4,  and  3,  we  find  3*,  4", 
53,  or,  which  is  the  same,  729,  256, 125.  Taking  the  12th 
root  of  these  results,  they  become 

(729)^'^,  (256)^'^,  (125)^'^. 

1  ? 

2.  Reduce  a^  and  x^  to  surds  expressing  the  same  root. 

Ans.  )  ^    '^     J 

(  x^  =(ar*)«. 

3.  Reduce  x^,  y^,  m^  to  surds  exprt?  sing  the  same  root 

x'^  =  {x')', 

Ans.;^!^^^.)^^ 

1  i 

4.  Reduce  V2,  V3,  V4  to  surds  expressing  the  same 
root. 

'  V2  =  (220)«'°, 
V4=:(4'=^)«''. 


(•A:M-    Iir 


(111.)  To  reduce  surds  to  their  simplest  form.  When 
ever  a  surd  can  be  separated  into  two  factors,  one  of  which 
is  a  perfect  power,  it  can  be  simplified  by  this 


144 


SURD   QUANTITIES, 


RULE. 

Having  separated  the  surd  into  two  factors,  one  of  which 
is  a  perfect  power,  take  the  root  of  the  factor  which  is  a 
perfect  power,  and  multiply  it  by  the  surd  of  the  other  fac- 
tor. 

EXAMPLES. 

1.  Reduce  v^SSS  to  its  simplest  form. 

We  can  separate  288  into  the  factors  144X2,  of  which 
144  is  a  perfect  square  who=e  root  is  12  ;  therefore 

v/288=  v/l44x2=  n'mIx  v/2  =  12v/2. 

2.  Reduce  Vx^y  —  a-x-  to  its  simplest  form. 

Ans.    Vx'-y  —  a-x'  =  x  '^y  —  a?. 

3.  Reduce  V—  32a*6  to  its  simplest  form. 


Ans.   V— 32a^6  =  — 2a  V6. 


4.  Reduce  (a-x^i/~')'  to  its  simplest  form. 


Ans.   (c-x^y-')    =iax^y-'--. 
5.  Reduce  (?7i^nx^y^)    to  its  simplest  form. 


f 


Ans.  mxy{nxy) 


(112.)  When  a  surd  is  in  the  form  of  a  fraction,  it  may 
be  simplified  by  the  following 


RULE. 

Multiply  both  numerator  and  denominator  by  such  a  quan-' 
tity  as  will  render  the  denominator  a  perfect  power. 


SURD    QUANTITIES.  145 


Reduce    -i/  —  to  its  simplest  form. 


Multiplying  both  numerator  and  denominator  by  11,  we 
have 


2.  Reduce    \/  — 7  to  its  simplest  form. 


3        /-T—       3 


1 

3.  Reduce  (— )  to  its  simplest  form, 

\xy  I       xy 

(_1L— 2v  3 
J  to  its  simplest  form 

ADDITION    AND    SUBTRACTION    OF    SURDS. 

RULE. 

(113.)  Reduce  the  surds  to  their  simplest  form ;  theii^if 
the  surd  part  is  the  same  in  both,  add  or  subtract  the  ration 
al  parts,  and  annex  the  common  surd  part  to  the  result; 
hut  when  the  surd  parts  are  different,  they  can  only  be  ad- 
ded or  subtracted  by  the  aid  of  the  signs  -|-  or  — . 
19 


146 


SIRU    QIjANTITIES. 
EXAMPLES. 


1.  What  is  the.  sum  of  v/54  and  \/24  ?     Also,  what  is 
the  difference  of  the  same  surds  ? 

By  reduction  we  have 

v/  54  =  ^9X6  =  3  x/  6 
,/  54  =  v/^x6  =  2  v/  6 


Therefore,         v/54 -fv/ 24  =5v/6. 

And,  v/54  —  v/  24  =     ^  6. 

2.  What  is  the  sum  and  difference  of  \/a^b  and  X/ab*  1 

The  sum  ={a  4-  h\i/ah. 

The  diff  =(a  — ?))Vfl6. 


Ans. 


3.  What  is  the  sum  of  (36x=^)'  and  (252/)'  ^■ 

Ans.  (6x  -|-  5)s/y. 

4.  What  is  the    .  i    of  (8a:)^  {xy')  ^  and  i21x*f  7 

Ans.  (2  4-3x4-3/'-')Va:. 

5.  What  is  the  sum  of  (a6V)'  and  {m.y)^  1 

Ans.  a:(a6"a:)^-j- 3/-(m^)*. 

MULTIPLICATION    AND    DIVISION    OF    SURDS, 

RULE. 

(114.)  Reduce  the  surds  to  equivalent  ones  expressing  the 
same  root,  (Case  II.  Art.  110,)  then  multiply  or  divide  as 
required. 

EXAMPLES. 

1.  What  is  the  product  of  v/8  by  V16  ? 


By  Case  11,  we  find 


Therefore, 


^8  =  (8-^)*=(512)*. 
V16  =  (16=^)^=(256)*. 
v/8x  V16  =  (512X256)*=4V32. 


SURD    QUANTITIES.  147 


2.  What  is  the  product  of  4Vfl6  by  3^/by 


I  7 


are  binomial  surds. 


Ans.  12Va-6y. 

3.  Divide  4  V32  by  V16. 

Ans.   v/8. 

4.  Divide  "^a'-b-^  by   ^aM. 

5.  Divide  v^4tt"6-^  by  ^2^2^ 

Ans.  a{ieab^y. 

EXTRACTION    OF    THE     SQUARE    ROOT    OF    A    BINOMIAL     SURD. 

(115.)  When  one  or  both  of  the  terms  of  a  binomial  are 
surds,  it  is  called  a  binomial  surd. 
Thus, 

a±Vb 
Vc±Vd 

(116.)  Before  we  proceed  to  the  extraction  of  the  square 
root  of  a  binomial  surd,  we  will  establish  the  following 
le7nmas. 

LEMMA    I. 

The  square  root  of  a  rational  quantity  can  not  consist 
of  the  Sinn  of  five  parts^  one  of  which  is  rational  and  the 
other  irrational. 

For  if  possible,  suppose  we  have  the  relation 

s/a  =  x-\-  v/y,  (1) 

where  z  is  rational. 

Squaring  both  members  of  (1),  we  find 

a  =  3ca-h2x^/y-f  y  (2) 

From  (2)  we  obtain 

a  —  x* — y  /o\ 

^/y  =  ^^.  (3) 


US. 


SrUD    EQUATIONS. 


Equation  (3)  gives  an  irrational  quantity  equal  to  a  ra- 
tional one,  Avhieh  is  impossible,  therefore  the  condition  (1) 
is  impossible,  hence  the  above  lemma  must  be  correct. 

LEMMA    II. 

In  any  equation,  cousistincr  of  rational  quantities  and  ir- 
rational quantities,  the  rational  quantities  on  each  side  are 
equal,  <!S  aho  are  the  h  rational  quantities. 

Suppose  we  have  the  equation 

a  -H  v/6  =  X  -I-  v/y.  (1) 

Then  if  a  is  not  equal  to  x,  let  us  have  o  =  x  ±  wi,  this 
value  of  a  substituted  in  (1),  gives 

x±:7n-\-  Vh  =  x+  Vy-,  (2) 

or         im-j-  -y/i=:  v/j/.  (3) 

Equation  (3)  shows  that  the  square  root  of  y  is  partly 
rational  and  partly  irrational,  which  is  impossible  (Lemma 
I).  Therefore  it  is  absurb  to  suppose  that  .r  differs  in  value 
from  0,  hence  x  ^  a.  Consequently  y/b  =  ^/y.  So  that 
the  above  lemma  is  correct. 


LEMMA    III 


// 


a  -\-  y/b  =  X  -{-  y/y,  then  icill  ^ a  —  ^/ft  =^x  —  y/y. 


If  we  square  both  the  members  of  the  equation 


find 


Va  -|-  v/ft  =  x-f-  v/y 


(1) 


a  +  v/6  =  a;-+  2x  y/y  +  y.  (2) 

Equating  the  rational,  as  well  as  the  irrational  parts  of 
(2),  (Lemma  II),  we  have 

a  =  x»+  y,  (3) 

y/b  =  2xVy-  (4) 


SURD    QUANTITIES.  149 

Subtracting  (4)  from  (3),  we  have 

a — ^b  =  x-  —  2xVy-\-y-  (5) 

Extracting  the  square  root  of  both  members  of  (5),  we  get 


Va  —  Vb  =  x-Vy.  (6) 

So  that  if  (1)  is  true,  then  also  will  (6)  be  true,  which  es- 
tablishes the  above  lemma. 

(117.)  We  are  now  prepared  to  proceed  to  the  extraction 
of  a  binomial  surd. 

Assume 


Va-{-Vb  =  x-{-Vy.  (1) 

Then,  (Lemma  III) 


^a  —  Vb=x—Vy-  (2) 

Equations  (1)  and  (2),  when  squared,  become 

a-\-^b  =  x^-\-2xVy-\-y.  (3) 

a  —  ^b=x'^  —  2x^y-\-y.  f4) 

Taking  the  sum  of  (3)  and  (4),  and  dividing  the  result  by  2, 
we  obtain 

a=x2-f-y.  (5) 

If  we  multiply  together  equations  (1)  and  (2),  we  get 


Va''  —  h  =  x^  —  y.  (6) 

Adding  (5)  and  (6),  and  dividing  by  2,  we  have 


a±Va^-b^ 

2  ^  ' 

Subtracting  (6)  from  (5),  and  dividing  by  2,  we  find 

a  —  v/a' — b  /Q\ 

2 =y.  (^) 

Extracting  the  square  root  of  both  members  of  (7)  and  (S), 
we  get 


150 


y/y- 


SURD    QUANTITIES. 

(9) 

_)  a—Va'^  —  b  I 

(10) 

Taking  the  sum  of  (9)  and  (10) ,  we  have 


\a-{-^a^  —  b(    j^)a—Va^  —  b(    ^     ^^^ 


x-j-vy- 

Subtracting  (10)  from  (9)^  we  get 


Va-_6 


(12) 


For  the  left-hand  members  of  (11)  and  (12) ,  substitute  theii 
vahies  given  by  (1)  and  (2),  and  we  then  have 


v^a+vfe 


=  )  a-{-^a^  —  b  I    -^\  a—^^a'  —  b 


(A) 


2 


—  Va'^  —  b 


(B) 


By  using  the  double  sign  dcj  we  may  combine  in  one  for- 
mula, both  (A)  and  (B). 

h 


v/.":r;7r=W  +  ^°^-M    ±)a-^a'-b(    .(C) 
'(  2)1  2  \ 

(118.)  We  will  now  show  the  use  of  formulas  (A)  and 
(B)  by  the  following 


SURD    QUANTITIES.  151 


EXAMPLES. 


1.  What  is  the  square  root  of  7  —  2v/10  / 
Reducing  the  factor  2,  to  the  form  of  the  square  root 
(Art.  109),  and  then  introducing  it  under  the  radical  sign, 
we  have  7  — 2  v/10  =  7  —  x/40,  which,  when  compared 
with  the  general  form  a—Vb,  gives  a  =  7  ;  &  =  40,  these 
values  of  a  and  &,  substituted  in  (B),  give 


,__V^=l  )  ^  (  7-v^49-40  )      /LzL^jL^  2. 

Therefore,  we  have 

v/7Tr27rO  =v/5*— v/2. 

2.  What  is  the  square  root  of  6  +v/20  ? 
[n  this  example  we  have  a  =  6  ;  6  =  20,  which  substi- 
tuted in  formula  (A) ,  gives 


v/6  -|-v/20=v/5  +  1. 

3.     What  is  the  square  root  of  2{x  -f-  1)  +  4  v/J-  ? 

Ans.   v/2x+s/2. 


2 

r 

(6+V/36- 

-20 

a  —Va''—  b 

U— v/36 

—  20 

2 
Therefore, 

1              ^ 

152 


SURD    QUANTITIES. 


4.  What  is  the  square  root  of  6  —  2>yo  ? 

Ans.    v/5  —  1. 

5,  What  is  the  square  root  of  7  -|-  4s/3  ? 

Ans.  2  +v/3. 

TO    FIND     MULTIPLIERS    WHICH    WILL    CAUSE     SURDS    TO 
BECOME    RATIONAL. 

CASE   I. 

(119.)  When  the  surd  consists  of  but  one  term,  we  can 
proceed  as  follow^s  : 

1  rn—l 

Suppose  the  given  surd  is  x"\  if  we  multiply  this  by  a;  m  * 

1  m—l 

by  rule  under  Art.  114,  we  shall  have  x  "'Xx   m  =x,  a  ra- 
tional quantity. 

Hence,  to  cause  a  monomial  surd  to  become  rational  by 
multiplication,  we  have  this 

RULE. 

Multiply  the  surd  by  the  same  quantity^  having  such  an 
exponent,  as  when  added  to  the  exponent  of  the  given  surdj 
shall  make  a  unit. 

EXAMPLES. 


1.  How  can  the  surd  x^  be  made  rational  by  multiplica- 
tion. 

In  this  example,  §  added  to  the  exponent  i,  gives  1, 

3 

therefore  we  must  multiply  by  x%  performing  the  operation, 
we  have 

1  2 

x^Xx  ^=x. 

3 

2.  Multiply  x^  so  that  it  shall  become  rational. 

3         a 

Ans.  x*Xa:*=a^ 


SUKD    QUANTITIES.  153 

3.  Multiply  x~^  so  that  it  shall  become  rational. 

4  11 

Ans.  x~''  Xx  "^  =x. 
CASE    II. 

(120.)  Wh^n  the  surd  consists  of  two  terms,  or  is  a  bino- 
mial surd. 

Suppose  it  is  required  to  multiply  -/«  +  \/&  so  as  to  pro- 
duce a  rational  product  ;  we  know  from  Art.  35,  Theorem 
III,  that 

(  v/a  +  \/6)  X  ( s/rt  —  v/6)  =  a  —  h. 
Hence,  to  cause  a  binomial  surd  to  become  rational  by 
multiplication,  we  have  this 

RULE. 

Change  the  sign  which  connects  the  two  terms  of  the  bi- 
nomial  surd,  from  -j-  to  — ,  or  from  —  to  -j-,  and  this  re- 
sult, multiplied  by  the  binomial  surd,  mil  give  a  rational 
product. 

EXAMPLES, 

1.  Multiply  v/3  — s/2  so  as  to  obtain  a  rational  product. 

Ans.  (v/3— v/2)X( v/3-(-v/2)^  3  —  2  =  1. 

2.  Multiply  4-f-\/5  so  that  the  result  shall  be  rational. 

Ans.  (4+^/5)X(4— v/5)  =  n. 


3.  How  canv^a-{-6 —  ^a  —  b  be  made  rational  by  mul- 
tiplication ? 

Ans.  (  v/a+6  —  Va  —  b)  X  (  V^b^  ^a  —  b)=  2*- 

4.  How  can  y/l  —  1  become  rational  by  multiplication  ^ 

Ans.  (s/7  — l)x(v/7  +  l)=:6. 


154 


SURD   QUANTITIES. 


(121.)  If  the  surd  consist  of  three  or  more  terms  of  the 
square  root,  connected  by  the  signs  plus  and  minus,  it  can 
be  made  rational,  by  first  multiplying  it  by  itself  after  chang- 
ing one  or  more  of  the  connecting  signs. 

EXAMPLES. 

1.  If  it  is  required  to  make  v/5 — v'3+i^2  rational  by 
multiplication,  we  should  first  multiply  by  -v/5  +  v/3-}--v/2j 
by  which  means  we  obtain 

^/5—    v/3+      ^/2 

v/5-f    ,/3+      v/2 


5— V/15+    ^10  — 3+N/6 
-j-v/15+    v/10         — v/6+2 


5  4-2\/10  — 3  +2=2v/10-|-4 

Again,  multiplying  2  x/lO  +  4  by  2\/  10  —  4,  we  get 
(2v/10  +  4)  X  (2v/10  —  4)  =  24. 

2.  Multiply  2  4-\/3 — v/2  so  that  it  shall  become  ra 
tional. 

FIRST    OPERATION. 

2  +    ^3—    v/2 
2  +    v/3  4-    ^-2 


4+2v/3  — 2^2  +  3— v/6 
+  2^3-1-272  +^6  —  2 

4-f4^/3  -f3  — 2=4y3-f5. 

SECOND    OPERATION. 
4v/3  +5 

4^/3—5 


48 


+  20v^3 

—  20v/3  — 25 


48 


—  25=23. 


SURD    QUANTITIES.  155 

3.  Multiply  v/5  +  72  —  y3  +  1  so  that  its  product  shall 
be  rational. 

FIRST    OPERATION. 

v/5  +    v'S—  v/3+    1 
V5  —  v/2  +    v/3  +    1 

~5~+^10— V/15+     ^/5+    ^/6— ^/2+n/3 
_2— yl0+yl5+    n/5  +    v/6+v/2— v/3 

3 

1 


1  +2^/5+2^/6. 

SECOND    OPERATION. 

l+2v/5+2v/6 
1—2/5  +2v/6 

14_2y5  +2^/6— 4v/30 
_20  — 2v/5+2v/6  -h4v/30 
24 

5  +4V6. 

THIRD    OPERATION. 
5-[-     4V6 

_5^   4V6 

—  25  —  20v6 
96  4-  20v6 

^    71. 
(122.)  To  reduce  fractions,  having  polynomial  surds  for 
a  numerator  or  denominator  or  both,  so  that  either  the  nu- 
merator or  denominator  may  be  free  from  radicals. 
Suppose  we  wish  to  transform  the  fraction 

1_ 

V3  +  V2  +  1  ' 
into  an  equivalent  fraction,  having  a  rational  denominator. 


156 


SURD    QUANTITIES. 


It  is  evident  that  this  transformation  can  be  eflfected,  pro- 
vided we  multiply  both  numerator  and  denominator  by  such 
a  quantity  as  will  cause  the  denominator  to  become  free  of 
radicals,  so  that  the  operation  is  reduced  to  the  finding  a  mul- 
tiplier which  will  make  v/3  +  ^/2  -[-  1  rational. 

We  will  first  multiply  by  —  ^  3  +  v/2  +  1. 

OPERATION. 

V3+V/2  +  1 
_^3  4-v/2  +  l 

—     3— v/6— y3-f-    v/2 

_|_     2-fv/6+^3+    ,/2  I 


2v/2. 
Hence,  if  we  multiply  both  numerator  and  denominator  ol 


by  —  v/3+  y/2-\-l  it  will  become 


1+^2— v/3 


y3H-v/2+l    J       ^      '   ^      '  2s/2 

Again,  multiplying  both   numerator   and  denominator  of 

— -!— ^^ ^—  by  y2,  we  finally  have —  .     The 

2^/2  -^         '  -^  4 

denominator  is  now  rational. 


(123.)  Hence,  to  transform  a  fraction,  ha^-ing  surds  in  its 
numerator  or  denominator  or  both,  into  an  equivalent  frac- 
tion, in  which  the  numerator  or  denominator  may  be  free  of 


RULE. 

Multiply  the  numerator  and  denominator  by  such  a  quan- 
tity as  will  cause  the  numerator  or  denominator,  as  the  re- 
quired case  may  be^  to  become  rational. 


SURD    QUANTITIES.  15" 


EXAMPLES. 


1.  Reduce  — ^^I^ —  to  a  fraction  havino-  a  rational  nu- 

4  ° 

merator. 

Multiplying  both  numerator  and  denominator  by  5  — V3, 
we  have 

5  +v/3_  (5 -f-s/3)(5  —  ^/3)_         22         _       11 


4  4(5 —v/3)  20  — 4v/3       10-2  x/3 

/g         j        O 

2.  Reduce  — r — ; — -, to  a  fraction  having  a  ration- 

al  denominator. 

Multiplying     both    numerator     and     denominator    by 
v/.5  -j-v/3  — v/2,  we  get 

5  +v/15  — y/lO  +  2^/5  +  2v/3— 2v/2 
6  -t-  2v/15  • 

Again,  multij)lying  both  numerator  and  t'enonihiator  of 
this  last  fraction,  by  C  —  ~\/15,  it  btroines 

4v/30  — 4^/15— 6^/10+10v/6— 8v/3—  12^/2 
—  24  ' 

or  changing  the  signs  of  both  numerator  and  denominator, 
it  becomes,  after  striking  out  the  factor  2  from  each, 

6^/2  +  4v/3  —  5v/6  -f-3N/10  -f  2v/15  — 2^/30 
12 

3.  Reduce ^ —  to  an  equivalent  fraction  having 

1    -r-  \/Z 

a  rational  denominator. 

.         ^/14  — /l)  —  s^T -l-v/5 
Ans.    z ' . 


4.  Reduce 


SURD   QUANTITIES 
1 


y3  — n/2  -f  1 
having  a  rational  denominator. 


to    an    equivalent    fraction 


Ans. 


2  — v/2  -hx/6 


5.  Reduce  — p-— ^ ,  first  to  a  fraction  having  a  ration- 

al  denominator,  and  then  to  a  fraction  having  a  rational 
numerator. 


Ans. 


■v/6  -^-Vx  b  —  X 


y/b — y/x       y/ab -^\/ax -\-y/bx  —  X 


IMAGINARY    QUANTITIES.  259 


IMAGINARY  QUANTITIES. 

(124.)  We  have  already  shown,  that  (see  Note  to  the 
Rule  under  Art.  96,)  an  even  root  of  a  negative  quantity  is 
impossible.     Such  expressions  are  called  imaginary. 

^~^'  J 

are  all  imaginary  quantitie*:. 


Surd  quantities,  though  their  values  can  not  be  accurately 
found,  can,  nevertheless  be  approximately  obtained  j  but 
imaginary  quantities  can  not  have  their  values  expressed  by 
any  means,  either  accurately  or  approximately.  They  must, 
therefore,  be  regarded  merely  as  symbolical  expressions. 

(125.)  We  will  confine  ourselves  to  the  imaginary  ex- 
pressions arising  from  taking  the  square  root  of  a  negative 
quantity. 

The  general  form  of  imaginaries  of  this  kind,  is 


^ —  a  =v«x —  1  =yaX  -^ —  1, 
substituting  h  for  Va,  we  have 

so  that  all  imaginary  quantities  arising  from  extracting  the 
square  root  of  a  minus  quantity  are  of  the  form 


160 


IMAGINARY    QUANTITIES. 


(126.)  If  we  put  v^ —  1  =z  c,  we  shall  always  have 
c-'=  —  1, 

c^        1, 


V-l. 


And  in  general. 


=  1, 


c4m  +  2^  _   1, 

m  being  any  positive  integer  whatever. 

(]27.)  From  which  we  easily  de;duce  the  following  prin- 
ciples. 

1.  (  +  v/ir^)x(+v/^^)=— ^a^=  — a. 
3.  (+^/I^7)x(— >/^^^)  =+  v/ «'=  +  «• 

^-    (— ^/— "7)X(— ^/^^)  =  —Vab. 
6.   (-f  v/ir7)x(_v/^i)  =  -\-^ab. 

The  above  is  in  accordance  with  the  usual  rules  for  the 
multiplication  of  algebraic  quantities,  and  must  be  consid- 
ered as  a  definition  of  this  symbol,  and  of  the  method  of 
using  it,  and  not  as  a  demonstration  of  its  properties. 

(128.)  The  student  must  not  infer  from  what  has  been 
said,  that  imaginary  quantities  are  useless.  So  far  from 
being  useless,  they  have  lent  their  aid  in  the  solution  of 
questions,  which  required  the  most  refined  and  delicate 
analysis. 

(129.)  We  will  now,  in  order  to  become  more  familiar  with 
the  operations  of  imaginaries,  perform  some   examples  in 


IMAGIXAKY    QIANTITIES.  jQ] 

MULTIPLICATION  CF  IMAGINARIES. 

1.  Multiply  4  v/ 131  j_  ^  IZ2  by  2v  iry_x/Zr3. 

OPERATION. 

2--'— 1  —  ^  -^ 
—  8— 2v/2+4v/3+v/G. 

2.  Multiply  4  +  v/ZTs  by  2  —  v/^2. 

OPERATION. 

4+    v^^^ 

2—     ^/^2 


8-f2v/— 3— 4v/— 2+v6. 
3.  Multiply  3  —  v/^^1  by  4  +  ^/^^ 

OPERATION. 

3—  ^/::^^ 

4+     v/=^ 


4.  Multiply 


12— 4n/— 1 

-|-3v/— 1  -f  1 

12—    %/- l-^l  =  l3_ 

V_l. 

.^  v^ —  3  into  itself. 

OPERATION. 

i  — i^^^3 

i-i^/~3 

^-^^^=^3 

-i^-3-l 

jV-3 

i_^x/_.3-5=_i_ 

21 

162  IMAGINARY    QUANTITIES. 

(130.)  Wc  will  now  ptMibrm  some  examples  in 

DIVISION  OF  IMAGINARY  QUANTITIES. 

1 .  Divide  4  +  ^'^2  by  2  —  ^^^. 

OPERATION. 

4-f  n/— 2 

Multiplying  numerator  and  denominator  by  2 -|-'^ — 2,  i* 
becomes 


=  1_^n/_2. 


In  the  same  way  we  find 

1  +  ^^- 


3. 


4. 


1-N/-1 

6n^^=^3 
2v/^^ 
3-v'~~l 


=  V. 


=  1^/3. 


4-f-2v/— 1 

(131.)  We  will  also  add  a  couple  of  examples  of  the  ex- 
traction of  the  square  root  of  imaginary  binomial  surds. 

1.  Extract  the  square  root  of  3  -J-2n/ —  1. 
Comparing  this  expression  with  the  general  formula  (A) 
Art.  117,  we  havo  a  =  3  :  b  =  —  4  :  hence, 


IMAGINARY    QUANTITIES.  \Ci?> 


*  .    {  v/13  — 3  )  ■     

■V— 1 


2.  Extract  the  square  root  of  3  —  2-^ —  1. 

All  the  difference  between  this  example  and  the  last,  is 
in  the  sign  which  connects  the  two  terms,  so  that  we  need 
only  change  the  sign  which  connects  the  two  terms  of  the 
answer  to  the  last,  in  order  1o  obtain  the  answer  of  this. 
(Compare  formuU.s  (A)  and  (B),  Art.  117.) 

Hence, 


V'       "  '       j         2  ^  ^  2 

If  we  add  the  cinswers  of  these  tvvo  questions,  we  shall 


ha 


ve 


0  +  2v/^i-y  3  -  2v/^  =2  j  ^^ii±^ 


=  v^2(v/13  -4-3). 
Jn  the  same  way  we  find 

(132.)  Before  closino;  this  chapter ,  we  unit  shoiv  the  in- 
Ur fr elation  of  the  jcllowing  syrahols    -,    — ,    -. 

We  know  frum  the  nature  of  multiplication,  that  0  niuh 
liplied  by  a  finite  quantity,  that  is,  0  repeated  a  finite  num- 
ber of  times,  must  still  remain  equal  0,  hence  we  have  this 

condition 

0X^  =  0.  (1) 

Dividing  both  members  of  (1)  by  ^,  we  find 

0=1  (^) 


]G4 


SYMBOLICAL    EXPRESSIONS. 


Therefore  the  symbol  --  will  always  be  equal  to  0,  as 

long  as  .7  is  a  finite  quantity. 

(133- )  Since  the  quotient  arising  from  dividing  one 
number  by  another  becomes  greater  in  proportion  as  the 
divisor  is  diminished,  it  follows  that  when  the  divisor  be- 
comes less  than  any  assignable  quantity,  then  the  quotient 
will  exceed  any  assignable  quantity.     Hence,  it  is  usual  for 

a 

mathematicians  to  say,  that  —  is  the  representation  of  an 
infinite  quantity.  The  symbol  employed  to  represent  infi- 
nity is  QQ,  so  that  we  have 

i^CC-  (3) 

(134.)    Dividing  both  members  of  (1)  by  0,  we  find 
0 


A. 


(4) 


This  being  true  for  all  values  of  A  shows  that  -  is  the 
symbol  of  an  indeterminate  quantity. 

To  illustrate  this  last  symbol,  we  will  take  several  exam- 
ples. 

1.  What  is  the  value  of  the  fraction -,  when  x:=a  1 

ox — ab 

Substituting  a  for  a:,  our  fraction  will  become 

I = ::=-  =1071  indeterminate  qxiantity. 

hx  —  ah       ah  —  ah       0 

If,  before  substituting  a  for  r,  we  divide  both  numerator 
and  denominator  of  the  given  fraction  by  x  —  a,  (Art.  55,) 
we  find 

X- —  a-        X  -\-  a 
hx  —  ah  h 


SYMBOLICAL    EXPRESSIONS.  16.: 

Now,  substituting  a  for  x,  in  this  reilucctl  form,  we  find 
x-\-  a       a  -j-  a 2a 


Therefore,  —  is  the  true  value  of  --,  when  x.^a. 

b  ox  —  ab 


y;2  QJ^ 

2.  What  is  the  value  of ,  when  x  =  a? 

X-  —  2(ix  -\-  a- 

Writing  a  for  x,  we  fiml 

x^  —  ax  a-  —a^        0 

XT  —  2ax  +  a-       a-  —  2a-  +  a-       0 

If  we  reduce  this  fraction  by  dividing  both  numerator  and 
denominator  by  x  —  o,  we  find 


x-  —  2ax  -\-  a-       x  —  a 
Now,  writing  a  for  x,  in  the  reduced  form,  we  find 

-!—=-^=:'=(X.     (Art.  133.) 
X  —  a       a  —  a       U 

3.  What  is  the  the  value  of ; — ^^ ; ,  when  x=a? 

bx  —  ab 

When  a  is  substituted  for  x,  we  have 

3^—3aj^-\-3a-x  —  a'_a'  —  3a^-\-3a^  —  a^  _  0 
bx  —  ab  ab  —  ab  0 

Reducing  by  dividing  numerator  and  denominator  by  x  —  a, 
we  find 

x^  —  Sox^+Sa^x  —  a^  _ x-  —  2ox-l-rt- 
bx  —  ab  h 

Writing  a  for  x,  wc  have 

±=i"f±f.'='i!z^f!±i'  =  °  =  0.     (Art.  132.) 
6  4  6^^ 


I6G 


SYMBOLICAL    EXPRESSIONS. 


(135.)    From  the  above,  we  conclude  that  whenever  an 
algebraic  fraction  is  reduced  to  the  form  -,  there  exists  a 


factor  becomes  zero  for  the  particular  value  of  the  unknown 
quantity  made  use  of.  In  the  foregoing  examples  there  was 
very  little  difficulty  in  discovering  this  factor. 

It  is  obvious  that  examples  of  this  kind  may  be  chosen 
where  it  would  be  more  difficult  to  find  this  factor. 

In  the  fraction, 


a.  we  shall  have  for  its  value 


In  this  case  we  do 


not  readily  discover  the  factor  required  j  but  if  we  multiply 
the  numerator  and  denominator  each  by  v^|(a-  +  x^)  -j-x, 

it  will  become 

^{(I'  —  X^) 

We  now  discover  that  the  factor  sought  is  a  —  x.     Divi- 
ding numerator  and  denominator  each  by  a  —  x,  it  becomes 

^c-l-rr) 


Now,  when 


v'^(a-(-  x')-{-x' 
a,  this  last  expression  will  become 


Hence,  we  conclude  that  indeterminate  expressions  of  the 
above  kind,  when  properly  reduced,  will  take  one  of  the 
following  form?. 

=  a  finite  quantity. 


B 


no  value. 


—  =00=  aw  infinite  quantity. 


QUADRATIC    EQUATIONS.  16' 


CHAPTER  V 


QUADRATIC   EQUATIONS. 

(136.)  We  have  already  (Art.  66),  defined  a  quadratic 
equation^  to  be  an  equation  in  which  the  unknown  quantity 
does  not  exceed  the  second  degree. 

The  most  general  form  of  a  quadratic  equation  of  one 
unicnown  quantity,  is 

fix'-+  hx-=  c.  (1) 

Dividing  all  the  terms  of  (1)  by  a,  (Axiome  IV,)  we  find 

x-^+^r  =  -^  (2) 

a  a 

where,  if  we  assume  ^=  -,  and  S  =  -,  we  shall  have 
a  a 

x'^-\-^x=B  (3) 

Equation  (3)  is  as  general  a  form  for  quadratics  as  equa- 
tion (1). 

In  (3),  ^  and  B  can  have  any  values  either  positive  or 
negative. 

(137.)  When  ^  —  0,  equation  (3)  will  become 

x^  =  B,  (-1) 

which  is  called  an  incomplete  quadratic  equation,  since  om 
of  the  terms  in  the  general  forms  (1)  and  (3)  is  wanting. 


168 


QUADRATIC    EQUATIONS. 


(138.)  When  B  =  0,  equation  (3)  will  become 
X'  -\-  Jix  =  0, 

which  divided  by  x  is  reduced  to 

x-^Ji=0, 
which  is  no  longer  a  quadratic  equation,  but  a  simple  equa- 
tion. 

(139.)  If  ^^  =:  0  and  j5  =  0  at  the  same  time,  equation 
(3)  will  become 

x^=0, 
which  can  only  be  satisfied  by  taking  a;=  0, 

INCOMPLETE    QUADRATIC    EQUATIONS. 

(140.)  We  have  just  seen  that  the  general  form  of  an 
incomplete  quadratic  equation  is 


=  B. 


(1) 


If  we  extract  the  square  root  of  both  members  of  this 
equation,  we  shall  (Art.  96,)  have 

x  =  dzVB.  (a) 

Equation  (a)  may  be  regarded  as  a  general  solution  of 
incomplete  quadratic  equations. 

(141.)  To  find  the  value  of  the  unknown,  when  the 
equation  which  involves  it,  leads  to  an  incomplete  quadratic 
equation,  we  have  this 


RULE. 

I.  Clear  the  equation  of  fractions  by  the  same  rule  as  for 
simple  equations.     (Art.  70.) 

II.  Tlicn  transpose  and  unite  the  like  terms.,  if  necessary., 
observing  the  rule  under  Art.  73,  and  loe  shall  thus  obtain^ 
after  dividing  by  the  coefficient  of  x-,  an  equation  of  the 


QUADRATIC  EQUATIONS.  169 

form  of  a;-=  B.     Extracting  the  square  root  of  both  mem- 
bers, we  shall Jind  x=  z^^/B. 


1.  Given  ^±^+7  =  9,  to  find  a:. 

This,  \vhen  cleared  of  fractions,  by  multiplying  by  19, 
becomes 

a;' 4- 2  4-  133=171, 
transposing  and  uniting   terms,  we  find  x^=^  36.     If  we 
compare   this   w'ith   our   general  form,  we   shall  see  that 
B=^  36.     Extracting  the  square  root,  we  have  a:=:  it  6, 
or  as  it  may  be  better  expressed,  x  =  6  or  a:  =  —  6. 

or-  3       ,    1       346    ^     „    , 

2-Given  —  +-  =  ^g^,tofindx. 

This  cleared  of  fractions,  becomes 

147  -f-  343x'^=  346x2, 
transposing  and  uniting  terms  3x-'=  147, 

dividing  by  3  x^=    49, 

extracting  the  square  root,  we  find        x  =    ±7. 

3.  Given  x- -—  =  44,  to  find  x. 

do 


Ans.  x=:  ±  12. 


4.  Given  S  +  5x'-=  ^  +  4x-+  28,  to  find  x. 


Ans.  xz=  =h  5. 


X- 


5.  Given2+~— 7=1--|-13,  to  findx. 
o  J 

Ans.  x  =  =t  9. 
(142.)  We  must  be  careful  to  interpret  the  tlouble  sign 

:,  correctly,  the  meaning  of  which  is,  that  the  quantity 
22 


170 


QUADRATIC    EQUATIONS. 


before  which  it  is  placed  may  be  either  plus,  or  it  may  be 
minus.  It  does  not  mean  that  the  quantity  can  be  both 
plus  and  minus  at  the  same  time. 

(143.)  If  an  equation  involving  one  unknown  quantity 
can  be  reduced  to  the  form  a;"=J\'',  the  value  of  x  can  be 
found  by  simply  extracting  the  nth  root  of  both  members, 
thus, 

(144.)  Where  it  must  be  observed  (Art.  96.)  that  when 
n  is  an  even  number,  the  value  of  x  will  be  either  plus  or 
minus  for  all  positive  values  of  J\^,  but  for  negative  values 
of  JV  the  value  of  x  will  be  impossible.  When  n  is  an  odd 
number,  the  value  of  x  will  have  the  same  sign  as  ^A'^has. 

(145.)  If  the  equation  can  be  reduced  to  the  form  x"'=zJV^ 
then  X  can  be  found  by  raising  both  members  to  the  mXh. 
power,  thus  :  x=^.K"K  , 

(146.)  Where  x  will  be  positive  for  all  values  of  J\\  pro- 
vided m  is  an  even  number,  but  when  m  is  an  odd  number 
then  X  will  have  the  same  sign  as  JV. 

(147.)  Finally,  when  the  equation  can  be  reduced  to  the 
form  IL 

We  must  first  involve  both  members  to  the  mth  power, 
and  then  extract  the  ni\\  root,  or  else  we  may  first  extract 
the  nth  root,  and  then  involve  to  the  ??ith  power.     (Art.  98.) 

Thus, 

m 
X^=J^n. 


EXAMPLES. 

1.  Given = — -,  to  find  x. 

v/x  +    4       Vx  +    6 

This,  when  cleared  of  fractions,  becomes 


QUADKATIC  EQUATIONS.  171 

..  -r  34  v/o:  +  16S  =  j^  +  42  v/x  +  152, 
transposing  and  uniting  terms,  we  have 

8v/x  =  16, 
ilividing  by  8,  v'a:  =  2, 

raising  to  the  second  power,  x  =  4. 

2a 

2.  Given    y,' x  4- "^ a  4- x  ^=    , ,  to  find  x. 

vfl  -(-a; 

This  equation,  when  cleared  of  the  fractions,  by  multiply 
ing  by  '^ a  -\-  x,  becomes 

v'ax  -1-  x-  -{-  a  -\-  x  =  2a, 
v^aa;  -}-  x'^=a  —  x, 
squaring  both  members, 

ax  +  X-  =  a-  —  ■2ax-\-  a;'*, 
3aa:=  a~ 
a 

3.  Given  3  +  x^=  7,  to  find  x. 

Ans.  a:  =  -b8. 

4.  Given  {y" —  b)    =  a  —  rf,  to  find  y. 

Ans.  y=:{{a-dY+by'. 

5.  Given  ^x—^2  =  16  —  v/x,  to  find  x. 

Ans.  x  =  81. 

6.  Given  (x  -fa)    =  — -,  to  find  x. 

(x-a)* 


Ans.  x  =  rh(2a2  +  2a6  +  6= 


7 


7.  Given         ^ = ,  to  find  x. 

v/x  —  >/a:  —  a      ^      ^ 

.  «(lzbcr 


172 


QUADRATIC    EQUATIONS. 


8.  Given  ^^x+^x—  ^x—Vx  =  -\/—^ — .tofindx. 

2  V  X-\-^/X 


9.  Given 


Ans.  a;  =  — . 
16 

=  -— -,  to  nnd  X. 


1— N/l_a^i       1+v/l— x2        a;^ 


Ans.  a:=zb-. 
2 


COMPLETE  QUADRATIC  EQUATIONS. 

(148.)  We  have  already  seen,  that 

aa;-+fex=c,  (A) 

is  the  most  general  form  of  a  quadratic  equation,  vrhere 
a  =  the  coefficient  of  the  first  term  ; 
h  =  the  coefficient  of  the  second  term  ; 
c  =  the  term  independent  of  r. 
If  we  multiply  the  general  quadratic  equation  (A),  by  4fl, 
it  will  become 

4a2a;2 -f  4a6a;  =  4rtc.  (1) 

Adding  6^  to  both  members  of  (1),  it  becomes 

4aV  -I-  4,abx  -^b^=  b'  +  4ac.  (2) 

The  left-hand  member  of  this  equation  is  a  complete 
square,  equal  to  {2a:x-\-bY.  The  process  by  which  we  so 
transform  an  equation  as  to  cause  one  of  its  members  to  be- 
come a  complete  square,  is  called  Completing  the  Square. 
This  may  be  effected  by  the  following 

RULE. 

Let  the  quadratic  equation  be  reduced  to  this  form^ 
ax--\-bx  =  c.  Then  multiply  each  member  by  four  times 
the  coefficient  of  the  first  term.,  after  which  add  to  each 
member  the  square  of  the  coefficient  of  the  second  term. 


QUADRATIC  EQUATIONS.  173 


1.  Complete  the  square  of  the  equation  a;--|-3a?=  4. 
Multiplying  each  member  by  4,  we  have 

4:X'-\~  12a:  =  16. 
Adding  the  square  of  3  =  9,  to  each  member,  we  find 

4x^4-  12x-f-  9  =  25. 
The  left  hand  member  is  now  a  complete  square,  equal 
to  {2x  -f-  3)'^,  so  also  is  the  right  hand  member, 

2.  Complete  the  square  of  18x- — 3x  =  1. 
Multiplying  each  member  by  4  X  18  :=  72,  we  have 

1296a,-2—  216x  =  72. 
Adding  3-=  9,  to  each  member  we  finally  have 
1296x2—216x4-9  =  81, 
each  member  of  which  is  a  complete  square. 

3.  Complete  the  square  of  6x2 —  7x=  —  2. 

Ans,   144x^—  168x  +  49  =  1. 

4.  Complete  the  square  of  lOx^ —  99x=  10. 

Ans.  400x2—  3960x  -|-  9801  =  10201. 
Raving  completed  the  square  of  a  quadratic  equa 
tion,  if  we  extract  the  square  root  of  each  member,  the  result 
will  be  a  simple  equation,  but  as  the  square  root  of  a  quantity 
may  be  either  positive  or  negative,  it  follows  that  our  result 
will  be  equivalent  to  two  distinct  simple  equations.  Thus, 
returning  to  our  general  equation,  ax'-\-  bx  =  c,  which, 
when  its  square  was  completed,  became  4:a'-^x--\-  4abx-\-  b'= 
62-|~  4"C,  we  have,  by  extracting  the  square  root  of  each 
member, 


2ax-\-b=  ±:^b^-\-^ac. 
If  we  make  use  of  the  +  sign,  we  have 


2ax-f-6  =  v^62_|_4„(, 


174 


QUADRATIC    EQUATIONS. 


If  we  use  the  —  sign,  we  have 

2ax  -\-  6=  — ^/6■^-|-4ac. 

Hence,  a  quadratic  equation  must,  in  general,  yield  two 
distinct  values  for  the  unknown  quantity.  The  above  results 
give  at  once 


2a 
Uniting  these  values  by  the  aid  of  the  ambiguous  sign  i, 
which  is  read  plus  or  minun,  not  plus  and  minus,  we  have 


—  6d=^6'+4ac 
2^i  ■ 


(B) 


(149.)  This  may  be  regarded  as  a  general  solution  of  all 
quadratic  equations,  and  it  is  obvious  that  we  may  derive 
from  it  a  general  rule  which  will  apply  to  all  quadratic 
equations,  so  as  not  to  be  under  the  necessity  of  actually 
going  through  Avith  all  the  preliminary  steps  of  completing 
the  square.     The  following  is  such  a 

RULE. 

Having  reduced  the  equation  to  the  general  form  ax^-\- 
hx  ==  c,  we  can  find  x,  by  taking  the  coefficient  of  the  second 
term  with  its  sign  changed,  plus  or  minus  the  square  root 
of  the  square  of  the  coefficient  of  the  second  term  increased 
by  four  times  the  coefficient  of  the  first  term  into  the  term 
independent  of  x,  and  the  whole  divided  by  twice  the  coeffi- 
cient of  thefi/rst  term. 


35 y. 

1 .  Given  4x =  46,  to  find  the  values  of  x. 


QUADRATIC    EQUATIONS.  Yl', 

This,  when  cleared  of  fractions,  becomes 
4x=— 36+x  =  46x. 
Transposing  and  uniting  terms,  we  have 

4x2— 45x  =  36. 
This  compared  with  the  general  form 
ax^-{-  hx=c. 
gives      a  =  4;  6  =  —  45;  c  =  36. 
The  square  of  the  coefficient  of  the  second  term 
=  (—45)-=  2025. 
Four  times  the  coefficient  of  the  first  term  into  the  term 
independent  of  x, 

=  4X4X36  =  576. 
Therefore,  taking  the  square  root  of  the  square  of  the 
coefficient  of  the  second  terra  increased  by  four  times  the 
coefficient  of  the  first  term  in.o  the  term  independent  of 
T,  we  get 

rt=  v^2025  -f  576  =  ±  ^2601  =  dr  51 . 
This  added  to  the  coefficient  of  the  second  term  with  the 
sign  changed,  gives 

45  ±51, 
which  must  be  divided  by  twice  the  coefficient  of  the  first 
term.     Hence, 

45  rb  51 

If  we  take  the  upper  sign,  we  get 

.  =  1^+^=12. 

O 

If  we  take  the  lower  sign,  we  find 

45  —  51  3 


X 


S  4' 

3 


Therefore,  x  =  12,  or 

'  '  4 

Either  of  wliich  values  of  x,  will  verify  tie  cquUlon. 


176  QUADRATIC    EQUATIONS. 

2.  Given =9 ,  to  find  the  values  of  x. 

X  —  4  2 

This,  when  reduced  to  the  general  form,  becomes 

r»— 18a;=— 72. 

Squaring  18,  we  get 

(18)2=324. 
Four  times  the  first  coefficient  multiplied  into  —  72,  gives 

4X  — 72  =  — 288, 
which  added  to  3^4,  gives  36,  the  square  root  of  which  is 
±6. 

Therefore,  x  =  — - —  =  12  or  6. 


3.  Given  v/3x— 5  =     '^^'+^^^^  to  find  the  values  of  x. 

X 

Squaring  both  members,  we  have 

7x2-}-36x      7x  +  36 

3x  —  5  = . 

x^  X 

This,  cleared  of  fractions,  becomes 

3x2— 5x  =  7x4-36. 

Transposing  and  uniting  terms,  we  have 

3x2  — 12x=36. 

This  divided  by  3,  gives 

x2_4x=12. 


^^      -                4=b^/(4)2  +  4Xl2     4zb8     p   _      o 
Therefore,  x  = ^\ ~~2~~    '  ^^~^' 

3  3         27 

4.  Given 1 =^i  to  find  the  values  of  x. 

X* —  3x  '  x2-f-  4x     8x 

This,  by  reduction,  becomes 

9x=^— 7x=116. 


QUADRATIC    EQUATIONS.  177 


™,       ,               7±v^7='+4x  9X116     7±65     ,  ., 

Therefore,  x= ^■— =_^-=:4,  or— 3 J. 

X'-\- 12      X 
5.  Given  — 1— -  =  4x,  to  find  x. 

Tliis  reduced,  becomes 

x'  —  '7x=  —  l2. 


T,.       ,               7±^^72  +  4x— 12     7±1 
Therefore,  x  = ~ -= — - —  =  4,  or  o. 

(150.)  An  equation  of  the  form 

ax^-{-hx"  =  Cj  (A) 

can  be  solved  by  the  above  rule,  which  indeed  will  agree 
with  the  form  under  consideration  in  the  particular  case  of 
71  =  \. 

If,  in  the  above  equation,  we  write  y  for  a:",  and  conse- 
quently 3/'  for  x'^,  it  will  become 

which  is  precisely  of  the  form  of  (A),  Art.  148.  Conse- 
quently, 

"= i;r^- 

Re-substituting  x^  for  y,  we  have 


„_  — 6d=^fc^  +  4ac 
^  2^  ' 


This  value  of  x,  must  hold  for  all  values  of  the  constants 
n,  a,  6,  and  c,  whether  positive  or  negative,  integral  or 
fractional. 


23 


178 


QUADRATIC    EQUATIONS. 


EXAMPLES. 

1.  Given  ar*-]-ax^  =  6j  to  find  x. 

This  becomes  y'^-\-ay  =  6,  when  for  x*  we  write  y. 

—  a±  ^a\-\-4:b 
•'•3/  = H — =  ^' 


hence, 


—  a=t^a-+46 


'  2.  Given  Sx**  —  2a:"  =  8,  to  find  x. 

n         2  ±10  _ 

x^  =  -^  =  2, 


3.  Given2(l+a:  — x-)— ^l  +  x  — o:^  — -,  to  find 


If  for  1  -f-  ^  —  2;2,  we  put  y',  our  equation  will  become 
2f-y=-l, 
or 


I8y'-9y  =  -l, 
9=b3       1        1 

y  = 


36 


3'"' 6' 


hence 


2^=  9'°^' 3-6- 


Re-substituting  1+x  —  rr^,  for  j/^,  we  have,  when  we 
take  the  first  value  of  y", 

1 


1  +x — x^= 


9' 


9x2— 9x  =  8, 
9±3v/41       1   ,   1 


18 


:+^^41,or?-1^41. 


QUADRATIC    EQUATIONS.  179 

When  vre  take  the  other  value  of  y'^,  we  have 

or  36i'— 36t  =  35, 

36i24v'n       1    ,   1,,,       1       1  „, 
■■■'  = 72 =2  +  3^"'"2-3^"- 

Collecting  these  four  values  of  x,  we  find 
a:=i  — Av/41, 

4.  Given  ja;2 — -J     ~\-la" — ^1     = -,  to  find  the  \'ia- 

lues  of  X. 

This  equation  is  easily  put  under  this  form 


-  va-x* —  a*=x' vx* — a*. 

X  X 


This  squared,  becomes 

a*—  -  =  X*—  2ax  Vz'—a'^  a^x^—-. 
x'  ^  x-i 

By  transposing,  we  have 

x* — a* —  2aa:  Vx* —  a*-\-  a'^x^=  0. 

Extracting  the  square  root,  we  find 


>/a:' —  a* —  aa:  =  0, 
or  v^x^ —  a*=  ax. 

Squaring,  we  find 


a:^— a*=a2x2, 

or 

x*—a'^x^=  a\ 

Hence, 

2 

Consequently, 

180 


QUADRATIC    EQUATIONS. 


=  -f-^^-^/ 


(151.)  We  have  seen  that  the  general  form  of  a  quadra- 
tic equation,  ax'-j-  bx  =-  c,  gave,  for  the  value  of  the  un- 
known, the  following  expression  : 


—  h  ±v/62_L4ac 

a:= , 

2a 

When  a  =  1,  the  equation  ax~-\-hx  =  c,  becomes 

x'-\-bx=c.  (C) 

And  the  above  expression  for  the  unknown,  will  become 


—  &±v^6-+4( 


l-v^g+ 


(D) 


Now,  since  all  quadratic  equations  may  be  made  to  assume 
the  form  of  (C),  by  dividing  all  the  terms  by  the  coefficient 
of  a:-,  it  follows  that  formula  (D)  must,  when  properly 
translated  into  common  language,  give  a  general  rule  for  the 
solution  of  all  quadratic  equations.     The  following  is  the 

RULE. 

Having  reduced  the  equation  to  the  form  x^  -\-hx  =  c,we 
can  find  x  by  talcing  half  the  coefficient  of  the  second  term, 
with  its  sign  changed;  plus  or  minus  the  square  root  of  the 
square  of  the  half  of  the  coefficient  of  the  second  tertn  in- 
creased by  the  term  independent  of  x. 

EXAMPLES. 

1.  Givena:*—10x  =  — 24,  to  finder. 

In  this  example,  half  the  coefficient  of  the  second  term  is 
5,  which  squared  and  added  to  —  24,  the  terra  independent 
of  X,  is  1.     Extracting  the  square  root  of  1,  we  have  ±1. 

Therefore,  x  =  5  i  1  =  6,  or  4. 


QUADRATIC    EQUATIONS.  181 

X  7 

2.  Given  =- -,  to  find  x. 

a: +  60       3x  —  5 

This  cleared  of  fractions,  becomes 

3a;^  — 5a:z=7a:+420. 
Transposing  and  uniting  terms,  we  have 

3x-  —  I2x  =  420. 
Dividing  by  3,  we  have 

x2  — 4z=140, 
.  • .  a;  =  2  ±  12  =  14,  or  —  10. 

^    „.         x  +  12  ,        X  26  ^    .    , 

3.  Given =  — ,  to  find  x. 

X      ^x+12       5' 

Ans.  a;  =  3,  or  —  15. 

4.  Given  3x^  -}-  42x3  =  3321,  to  find  x. 

Ans.  3,  or  (—41)*. 

(152.)    Equations   containing  two  or  more  unknown 
quantities,  which  involve  in  their  solution 

quadratic  EQUATIONS. 


Given  <    ^        ,       Jl^^^     r  ?  to  find  x  and  y. 
I  y'i  — 7^  =  90000.     S 


From  the  first  of  these  equations,  we  find 
^__300y_ 
y— 125' 
Substituting  this  value  of  x  in  the  second  equation,  it  be 


y-_,J^r=  90000. 


y-125 
Which,  when  expanded,  is 


182 


QUADRATIC    EQUATIONS. 
90000/ 


=  90000. 


"^        /  —  2b0y  +  15625 
This,  cleared  of  fractions,  and  terms  united,  becomes 

yi  _  250 f  — 164375/  -\-  22500000^/  =  1406250000. 
This  may  be  written  as  follows 

(/  —  125y)-  —  180000(/—  125^/)  =  1406250000. 
Solving  by  rule  for  quadratics,  considering  /  —  125t/  as 
the  unknown  quantity,  we  have 


Hence, 


125y  =  90000  d=  97500. 


1251/ =  187500,  or  /—  125y=  — 7500. 


The  first  of  these  gives 

125  db  875 


:500,  or  —375. 


The  second  gives 


125  ±25x^—23 


Both  of  which  values  are  imaginary. 

Having  found  y,  we  can  substitute  it  in  the  equation 
_    Z00y_ 

^— ^pri25' 

and  thus  obtain  the  values  of  x. 


2. 

Given 

x^= 
-ex'' 

(1)? 
(2)5' 

to  find 

X  and 

y 

From  (2) 

,  we  get 

x* 

/- 

c' 

(3) 

bic 

h  substituted  in 

(1), 

we  have 

y'- 

ay 

—  a? 

(4) 

/- 

-2cy'4- 

c» 

QUADRATIC  EQUATIONS.  183 

Clearing  (4)  of  fractions,  it  may  then  be  put  under  the 
form 

(y6  _  cy^y.  —  2a-  {y"  —  cy^)=  a'-c-.         (5) 

Solving  this  by  quadratics,  considering  y^  —  cy^  as  the  un- 
known quantity,  we  have 

3/6  _  cy3  =  a2  dr  a  ^a*  -f  c-.  (6) 

Again,  solving  (6)  by  quadratics,  considering  y^  as  the  un- 
known, we  have 

Extracting  the  cube  root  of  (7),  it  becomes 

The  value  of  y,  (8),  or  better  the  value  oi  y^,  (7),  when 
substituted  in  (3),  will  give  x. 

v-\-w-{-x-^y-\-z=     56  (!)■ 

.vw  —  x  —  y  —  z=   207  (2) 

3.  Gi\'en<(wx  —  v  —  y  —  z=   —9  (3) 

\xy — V  —  w  —  z= — 19  (4)1 

yz — V — w — x=     38  (5) 


(8) 


,  to  find  V,  ty, 
ar,  y,  and  z. 


vw-\-v  +  w  =  263,        (6)=zz(l)-|-(2) 

y,x-{-w-{-x=   47,        (7)=(l)-f-(3) 

xy-{-x  +  y=   37,        (8)=(l)4-(4) 

yz^y-\-z=    94.        (9)=(l)+(5) 

By  adding  a  unit  to  both  members  of  equations  (6),  (7), 

(8),  (9),  they  may  be  put  under  the  following  forms  : 

(v -|-l)(wj+ 1)=264,        (10) 

(tx^-f  l)(x+l)=   48,        (11) 

(x+l)(y-f  1)=    38,        (12) 

(y  +  1)(2  +  1)=   95.        (13) 


184 


QUADRATIC   EQUATIONS. 


If  we  add  5  to  both  members  of  (1)  it  may  be  written 
as  follows  : 
(r+l)4-(u.-4-l)-f-(x+l)+(y+l)+(z+l)=61.     (14) 

We  shall  now  use  equations  (10),  (11),  (12),  (13)  and 
(14),  which  are  symmetrical  instead  of  the  original  equa- 
tions. 

264 


w-\-l 


v-\-r 

48         2 


(15)=(10)^(v  + 1) 


+  i=i;rqrr=fi^^+^^'    (i6)=(ii)-(t/;-fi) 

38         209 


3/+1 


+  1     v+l' 


(17)=(12)^(x+l) 


z^l=^=Uv-\-l).      (I8)=(13)--(y  +  l) 
y-f-l      11 

Substituting  these  values  oft«  +  l,  x-\-lj  y+l,  -  +  1) 
in  (14),  we  have 


264 


209 


This  reduces  to  this  form, 
18, 


^^;(.+  l)  +  iI3_61. 


(20) 


Clearing  of  fractions,  we  have 

I8(v+1)2  — 671  (v  +  l)=  — 5203.  (21) 

This  quadratic  solved,  gives 

v+l  =  11,  or  26/3. 

These  two  Values  of  v+  1>  ^^ing  substituted  in  (15) 

(16),  (17),  (18),  will  give  two  sets  of  values  for  w;  +  1 

x-^l,  y+lj  s+1.     These  values  when  found  are, 

v-f  1  =  11,  or26T\. 

M,-|-l=24,        10y»3. 


QUADRATIC    EQUATIONS. 


185 


a:  +  1  =    2,  or    41; 
y  + 1=^19,  or    7ii. 
z-f-l=    5,  or  lly^. 


Ay=25^ 

V  1^=23/ 

V'-=     ^^3 

h=    1,V 

or 

<^^=    31. 

;3/  =  isA 

)y=  611 

r=   4,  ) 

(  z  =  10|^ 

4.  Given  x'^"—2x^"-\-x"=.6,  to  find  a:. 
This  is  readily  put  under  this  form 

If  we  make  y  =  x^" — a;",  equation  (1)  will  become 

r—  2/  =  6,  (2) 

.••y  =  i±^  (3) 

Re-substituting  for  y,  we  have 

X-"— a.'"  =  3,  (4)  ) 

or  x-"—x''—--2.  (5)  \ 

Now,  in  (4)  and  (5)  substituting  c  for  a:",  and  we  have 

z'-z  =  3, 

From  (6),  we  have 

c=  >  ±  V13. 
From  (7),  we  find 

Re-substituting  x"  for  c,  we  find 

a:"=i±  ^n/13, 

a:"=  i  db  5  v^ — 7. 
Taking  the  7ith  roots  of  (10)  and  (11),  we  find 


(G)( 

(8)  I 
{9)S 


(10) 
(H) 


Ans 


24 


186  QUADRATIC  EQUATIONS 

5.  (jrnen  ^       '   -^  ^   ^ 


x^+r'=6  (2) 

Squaring  (1),  we  have 

x'-\-2xy-\-y''=a~. 
Subtracting  (2)  from  (3),  we  get 
2xy  =  a-  —  b. 
Subtracting  (4)  from  (2),  we  find 

■j^  —  2xy-\-y'  =  26  —  a-. 
Extracting  the  square  root  of  (5) ,  we  get 

x  —  y—±  N/2fe  —  a". 
Taking  half  the  sum  of  (1)  and  (6),  we  get 


to  find  X  and  y. 


a       1 


x  =  -:k-^2b  —  a^. 
2       2 

Subtracting  (7)  from  (1),  we  find 

(I        1    /— , :, 

•^       2       2 


(3) 
(4) 

(5) 

(6) 

(7) 

(8) 


(x  +  y  =  «  (1)} 

6.  Given  <        ,      ^       ,  >,  to  find  a;  and  y. 

(a-3-j-3/^'  =  6  (2)) 

We  will  indicate  our  operations  upon  the  successive  equa 
tions,  by  the  method  explained  under  Art.  80. 


'ixy[x-^y)  =  iv^  —  h. 

n-'  —  h 
3xy  — . 


a-y  = 


3a 


x^+2xy-\-y''  =  a- 


4a>  — 46 
^'^  =  —30-- 


(3)-(iy 
(4)=(3)-(2) 

(5)=(4)-(l) 
(6)=(5)--3 

(S)=  (6)X4 


QUADRATIC    EQUATIONS.  187 

4/) /j3 

^-2xyJry'  =  —^^.  (9)=(7)-  (8) 

(46  — flsH 


3c 


in)J-2Wm 


3a      \  •  '      '  2 


a       1     46-a3     i  (i)_(io) 


ix+y  =  a  (1) 

7,   Given  {  ^  1  If  find  x  and  v. 

ar^+4r^i/+6xV+4x3/3+y  =:a^  (3)=(1)^ 

4x1/  (a-2+/)+6a:y  =  a^-h.    (4)=:(3)— (2) 
a--^+2x3/+y-2=a2.  (5)=(1)^ 

Transposing  2xy  of  (5)\ve  get 

.x-^4-7/-^=:a-^— 2x3/.  (6) 

Substituting  this  value  of  x- -j- 3/*  in  (4)^  we  get 

4x7/(a2  —  2xy)  +6xV'  =  «'  —  ^-  C' ) 

This  becomes,  by  putting  z  for  xy,  and  transposing, 

2c2— 4a'r  =  6  — a',  (8) 

■.•.  =  «=±v/3^'.  (9) 

Hence, 

X2/  =  a-^±\/-+-.  (10) 

4x3/  =  4a^±4\/yii\  (Il)=(l0)x4 


188 


QUADRATIC    EQUATIONS. 


X  — y  =  =h 


=1^1    -3.=  T4/l±^«r.    (15)=W^) 


8.  Given 


(   x-  — y=  =  a         (1) 
Ix'y-^xf^h         (2) 


a^*  +  y-  =  -  • 

h     ,  a 
2xy^2 


to  find  a:,  and  y. 


{'i)={2)-rxy 


(4) 


__(3)+(l) 


2xy     2 


4a;Y     4 
This  read.ly  gives, 

4x'2/'  -{-  cL-x'-y-  =  b. 
Consequently, 


^   '  2 

(6)=(4)X(5) 


xy  =  ±  ^ — '. j 


/— rt^x/fl'-f-166^ 


(8) 


This  value  of  xy,  introiluced  into   (4)  and  (5),  we  obtain 
2 


x=  izl  doh 


^^/a^-\-16b!       2 


(9) 


QUADRATIC    EQUATIONS.  189 

-bl^ g V-^l^      (10) 

Cxy+a:V=135  (1)  ?     .    .    , 

x'Y  +2xV  +  xV"  =18225  (3)==(1)- 

2xY  =  13122  (4)=(3)_(2) 

a:Y  =  6561  (5)=(4)-^2 

xy=±3  (6)=-V(5) 

xY  =  ±:21  (7)=(6)3 

a:^-f-y^  =  ±5  (8)=r(l)-^(7) 

2x3/=  ±6  (9)=:(6)X2 

a:^-2x2/4-y^=  =F  1 (10)=(8)-(9) 

X— y=:±%/— lorrhl  (11)=>/(10) 

x-'  +  2x3/  +  y^=ill (12)=(8)+(9) 

x  +  7/  =  ±v/ll  orrbv^— 11  (13)=:v/(12) 

x=l{±Vll±V—^)or  i{±^^-n±i)  (i4)=^-^-ii^^ 

, ,     (13)— (11) 

i/=i(±:v/ll=Fv/— Dor  J(±v/-11=F1)  (lo)=^ ~ 

10    Given  ^^'^"'^'"^'^""^  ^\^  ?  .tofind  xandy, 

'''•'''''"  ^i/x(2/x-f-l)-x^4-x=6     (2)5' 

Subtracting  (1)  from  (2),  we  find 

y'^r^  -{-  yx  —  yx^  =  3.  (3) 

Dividing  (3)  by  (1),  we  get 

y=l.  (4) 

This  value  of  y  substituted  in  (1),  gives 
x=3. 
/xy— 8xV+16x^  N 

\     =90x.y+60(x-2/0-720(y-l)      (1)( 
11.  iiiven<  /  o      .      I  .X  lo  r  ■■ 

)  (r— 4y-H)x_^__12         ^2)V 

(  6  X  J 

to  find  X  and  y. 


190 


QUADRATIC   EQUATIONS. 


Multiplying  (2)  by  bx{y'^  +  4y  +  4),  it  becomes 

I  =15z^'^+60xy+60x— 60/— 240?/— 240  )  ^  ^ 

Subtracting  (1)  from  (3),  we  have 

0  =  Ibxy"  —  30x2/  +  48O3/  —  960.  (4) 

Dividing  (4)  by  Ibxy  +  480,  it  becomes 

0.=  y-2,  (6) 

.•.y  =  2.  (6) 

This  value  of  y  substituted  in  (2),  gives 

x=4.  (7) 

rxy+z=5  (1)^ 

12.  Given  ?  xyz  +  2'^=]5  (2)  >  5  to  find  x,  y 

(  xy*-j-x2y— 2x-i-2c=8      (3)  )     and  z. 
Dividing  (2)  by  (1),  we  find 

z  =  3.  (4) 

Substituting  this  value  of  z  in  (1)  and  (3)  and  they  be- 


xy  =  2. 

(5) 

xy(x+2/)  =  2  +  2x. 

(6) 

Dividing  (6)  by  (5),  we  find 

xH-y=l-f-x, 

(7) 

.'.  y=h 

(8) 

Dividing  (5)  by  (8),  we  get 

x=2. 

(9) 

rx{y+z)  =  a          (1)) 

13.  Given]  2/(2:+-)  =  &          (2)  > 

to  find  X 

, y, and  z 

(z{x-\-y)  =  c          (3)) 

Before  proceeding  to  the  solution  of  these  equations,  we 
will  remark,  that  they  are  symmetrica],  and  consequently 


QUADRATIC  EQUATIONS.  191 

all  the  derived  equations  will  either  contain  all  the  letters 
similarly  combined,  or  else  they  will  appear  in  systems  of 
three  equations  each,  which  can  be  deduced  from  each  other 
by  simply  permuting. 

If  we  take  the  sum  of  (1),  (2),  and  (3),  after  expand- 
ing them,  we  shall  have 

2xy-{-2yz-^2zx=:a-\-b~^c.  (4) 

In  this  equation  all  the  letters  enter  symmetrically  ; 
therefore  it  will  not  give  rise  to  any  new  equation  by  per- 
mutation. 

If  we  subtract  twice  (3)  from  (4),  we  get 

2xy  =  a-\-b-~c.  (5) 

By  permutation,  we  derive  from  (5)  these  two  equations : 
2yz  =  b-\-c-a.  (6) 

2zx=c-\-a  —  b.  (7) 

Equations  (5),  (6),  and  (7)  readily  give 

a-\~b  —  c 
^V  =  -^ (8) 

y-.^'j^.  (9) 

..  =  i±p.^  (10) 

Taking  the  continued  product  of  (8),  (9)  and  (10),  we 
have 

^a..a.i        \  a  +  6— c  )  ^^  (  b-\-c—a  )  ^^  (  c-\-a—b 


X    -V-    X    --.—   •    (11) 


(        ^         )        (        ^         )        (        2 

This  equation  containing  all  the  letters  symmetrically  com- 
bined, can  give  no  new  condition  by  permutation. 
Dividing  (11)  by  the  square  of  (9),  wc  have 
(a+6-c)(c4-a-6) 

^ 2{y^^ ^^^^ 


192 


QUADRATIC   EQUATIONS. 


By  permuting,  we  derive  from  (12)  these  two  equations  : 


r 


(64-c  — a)(a+6  — c) 


(13) 
(14) 


2(c+a  — &) 

^.-(c+^-&)(^+c-q) 
2(a+6  — c) 

Taking  the  square  roots  of  (12),  (13) ,  and  (14),  we  find 
(a^_6_c)(c+o  — 6)  )  i 


a;  =  rh 


y  =  ± 


2(6+c  — a) 
(6-fc  — fl)(a-|-6  — c) 


2  =  rb 


(15) 
(16) 
(17) 


2(c+a  — 6) 

(c+g  — 6)(6-fc  — g)  )  ^ 
^  2(g-f6-c)  ^    • 

This  question  is  a  good  illustration  of  the  beautiful  method 
of  deriving  one  quantity  from  another,  of  a  similar  nature, 
by  simply  permutating. 

14.  Given,  the  two  equations 
(x'-f  x")(l+x'a;"+a:'V'+x'a:"2+a:'-x"«)-fx'a:"  =  g, 
x'x"{x'  -^x"){x'-\-x"-\-x'x"){x'-\-x"-^x'x"+x'-'x"  -\'x'x"")  I 

to  find  x'  and  x". 

If,  in  these  equations,  we  make  successively  the  substitu- 
tions      x'4-  x"  =  3/',  x'x"  =  y" ;  y'+y"=  2',  y'y"=  z" ; 
z'^z"=w',  z'z"==w",  we  shall  finally  have 
w'-\-w"  =  0, 
w'w"  =  b. 

The  quantities  sought,  x',  x",  will  be  determined  by  means 
of  these  four  quadratic  equations  : 

w^  —  aw-\-b    =0. 
z  -^w'z-\-w"=0. 


QrADRATIC    EQIATIOXS.  193 

cc-  —  y'x  -^y"  =  0. 
The  first  of  tliesc  equations  determines  w'  iind  lo"  ;  the  se- 
cond  z'  and   z"  ;    the  third  y'  and  y"  ;    and,  finally,  the 
fouvth  x'  and  x" .     We  thus  suciessively  obtain 


a  ±  n/^;~- 

-46 

2 

_.„,'_ts/i«'^ 

—  4:10" 

2 

C'±v/C'^ 

-Az" 

2              ' 

w' 

(1  =FN^a-  —  46 

2 

^1 

_2o'=FN^w'-— 4t/;" 

2 

y' 

^_^'z:pVz'-^—4:z" 

2 

x' 

,_y^^y'— 4.v" 

,_iy'  ±^Vj— 4//" 
''-  2  '  "  2 

and  there  are,  consequently,  for  a:',  as  well  as  for  a;",  six- 
teen different  values.  If  we  had  solved  the  first  two  equa- 
tions by  the  common  method,  we  should,  after  a  laborious 
elimination,  have  obtained  an  equation  of  the  16th  degree. 
If  a  =  371,  and  b  =  13530,  then  will  one  set  of  values 
be,  x'  =  2  and  x"  =  3. 

Cx'+xy-]-y'=a',         (1)^ 
15.  Given  ^  y--\-yz  -f-  z*=6^,         (2)  >  to  find  a-,  y,  z. 
iz'-\-zx-\-x''=c\         (3)) 

=a2+62-f-c^ 


(4)=(l)  +  (2)+(3) 


4(x^-i-2/^-|-2^f+4(a:-^+2/^-|--^)(2-y+yz-fca:)       )  ,-._,, ^ 

-^{xy+yzJr~:ty^{a'  +  lr+c'y.  ^  ^^>'-^'*^ 

r'+3aV  +  2/-*  +  2a:'y-{-2y^x=:a'.  (6)=(1)^ 

y^+33/V  +  c'  +  2i/3z  +  2z^3/^6^.  (7)=(2)^ 

c-*  +  -iz^x'-  +  r'  +  2z^^a:  +  2x'»z  =  c\  (S)=(3)- 

irx'-j-y'^z'y-\^{T^-\-y^-^z%xy+yz-^zx)  )  (9):=2(6) 

-2(x3/+y;=+zx)^=2(aM-6'  +  c^).  i   +2(7)+2(8) 
25 


194 


QUADRATIC    EQUATIONS. 


(10)=(5)-(9) 


or,  which  is  the  same  thing, 

{:ri/+yz+zxf=?,{a:^h'^})'c'^c'a^)-^{a^+¥-\~c^).      (11) 
{xy-{-yz-{.zx)-= ) 

6(.T2/  +  7/c4-cx)-3:6/c.  (13)=(12)X6 


(14)=^4)X2 


4(x+.r/+c)^2K+6'^+c^)+6/c.  (15)=(14)+(13) 

2(a:+2/+c)=±  V2{a'-\-b''~\-c^6k.      ( 16)=v/(15J 
2(x^  +  r  +  -  +  x:v-Hz-f  c.)  1         .n)=(4)+(12) 

2x{x^y-i-z)  =  a'—b^-\-c'+/c.  (I8)=(17)-(2)x2 

x=  -^!=L±SL^.  (19)^(18)^(16) 

±  v'2(a2+62^c3)+6/c 

Having  found  the  value  of  x,  we  may  find  the  values  of  y 
and  c,  by  simply  permuting  the  letters  in  the  above  expres- 
sion, (19).  Since  the  expression  for  k  is  symmetrical,  i* 
must  remain  constantly  the  same.  Consequently  the  deno- 
minator of  the  expression  for  x,  (19),  will  not,  during  this 
permutation,  change  its  value. 

In  this  way  we  find 

y 


db  ^/2(a'-{-62-^-c•*)+6A:' 


(20) 
(21) 


QUADRATIC    EQUATIONS. 


195 


F 

"o 

C 

^^ ^ 

1+ 

,_^ 

+ 

I'- 

+ 

l"^ 

i-c 

<o 

j'-O 

r« 

■<5 

;  + 

+ 

+ 

1+ 

'4- 

,+ 

i^ 

's 

55 

j3 

-o 

1^ 

^    ;+ 

+ 

1    ,  ' 

+ 

« 

1  1 

— 

1 

_ 

1 

-=     « 

1 

e 

1 

e 

1 

—           1  'Z^^ 

1^ 

^ 

?7~" 

-In 

^ 

're         :  ,■ 

1 

c 

, 

1  '^ 

•^      :  1 

!  '^ 

1 

^ 

1 

1- 

i  + 

1« 

+ 

Js 

:+ 

\% 

■■^ 

^ 

'!? 

«o 

''■ , 

h 

4- 

:  + 

+ 

4- 

3.         i-^ 

i    jc      ■      ] 

-a 

~o 

Si 

Zi 

^     + 

+ 

'  'c 

+ 

^ 

\."      1 

■P 

^.     , 

<s 

^' 

CO              1 

CD               1 

^ 

4] 

o         V 

> 

r~^ 

> 

I       -H 

+ 

-H 

+ 

-H 

"5- 
+ 

f 

+ 

4- 

4- 

"O 

o 

e 

1 

'^ 

c: 

1 

^^ 

1 

1 

1 

^^ 

c 

5   ^ 

1 

3 

cT 

> 

« 

> 

4^ 

-H 

-H 

^^^^  QUAUIfATIC    KQLATIO.XS. 


^^  e  have  cho.en  this  example,  partly  from  its  being  one 
rather  .hfficult  of  solution  by  the  ordinary  methods,  and 
partly  because  it  affords  an  excellent  opportunity  for  exem- 
plifying the  beauty  of  syn.metrical  equations.     Equations 
U  ,  (2),  and  (3),  which  are  given,  are  not  only  symmetrica], 
but  they  are  also  homogeneous.     Consequently  all  our  de- 
rived equations  will  be  homogeneous,  and  will  either  con- 
tain all  the  different  letters  similarly  involved,  as  in  (4)  (51 
(9),  (10),  (11),  (12),  (13),  (14),  (15),  (16),  (17),  and  (IS), 
or  else  there  will  be  a  system  of  three  equations  which  can 
be  deduced  from  each  other  simply  by  permutating  the  let- 
tcrs,  as  IS  the  case  with  the  given  equations  (1),  (2),  and 
(3),  also  equations  (6),   (7),  and  (8).      Equations  (19), 
(20),  and  (21),  a,e  also  of  this  nature.     This  perfect  sym- 
"  etry  of  expressions,  must  in  a  great  measure  serve  as  a 
check  upon  our  work,  preventing  errors  which  otherwise 
could  not  be  so  readily  detected. 

(153.)  Questions  which  require  for  their  solution  a 

KNOWLEDGE  OF  QUADRATIC  EQUATIONS. 

1.  A  widow  possessed  13,000  dollars,  which  she  divided 
into  two  parts,  and  placed  them  at  interest,  in  such  a  man- 
ner that  the  incomes  from  them  were  equal.  If  she  had 
put  out  the  first  portion  at  the  same  rate  as  the  second, 
she  would  have  drawn  for  this  part  360  dollars  interest  j 
and  if  she  had  placed  the  second  out  at  the  same  rate  as 
the  first,  she  would  have  drawn  for  it  490  dollars  interest. 
What  were  the  two  rates  of  interest  1 

Let  X  =  the  rate  per  cent,  of  the  first  part. 

Let  3/  =  the  rate  per  cent,  of  the  second  part. 

Now,  since  the  incomes  from  the  two  parts  were  equal 


QUADRATIC  EQUATIONS.  197 

they  must  have  been  to  each  other  reciprocally  as  .r  to  y. 
Hence,  if  my  denote  the  first  part,  then  will  mx  denote  the 
second  part. 

We  shall  then  have 

„j  (:c  _]- y)=13000. 

13000 
Consequently,         m  =  '^JTy 

iSOOOy       ^,     r.    ,        , 
Therefore,  -^  =  t^^^  first  part. 

iSOOOo:        ,  , 

=  the  second  part. 

x-\-y 

The  interest  on  these  parts,  at  y  and  x  per  cent.,  respec- 
tively, is 

I30v-  ,  130x2 
— -^  and  — ; — . 
x-\-y  x-\-y 

Hpnce,  by  the  conditions  of  the  question,  we  have 

122^=360.  (1) 

X+J/ 

1^=490.  (2) 

x+y 

Dividing  (2)  by  (1),  we  get 

t-^1  .       (3) 

3/2-36- 

Extracting  the  square  root  of  (3),  we  have 

y     6 

Subtracting  (1)  from  (2),  we  have 

130(.2-/)^^3^  ^5^ 

x-\-y 
Dividing  boUi  numerator  and  denominator,  of  the  left-hand 


198 


QUADRATIC    EQUATIONS. 


member  of  (5),  by  a'+y,  and  also  dividing  both  members 
by  130,  we  get 


--r-2/=l. 

(6) 

Dividing  (6)  by  t/,  we  find 

^—1  =  1 

y           y 

(7) 

Subtracting  (7)  from  (4),  we  have 

-i-f 

(8) 

Clearing  (8)of  fractions,  we  obtain 

6y  =^1y  —  6. 

(9) 

•••  3/=6. 

(10) 

Adding  (10)  and  (G),  we  get 

x  =  7. 

(11) 

Therefore  the  per  cent,  of  the  first  part  w^as  7,  and  of  the 
second  part  was  6. 

2.  A  certain  capital  is  out  at  4  per  cent. ;  if  we  multiply 
the  number  of  dollars  in  the  capital,  by  the  number  of  dol- 
lars in  the  interest  for  5  months,  we  obtain  $117041|. 
What  is  the  capital  ?  Ans.  $2650. 

3.  There  are  two  numbers,  one  of  which  is  greater  than 
the  other  by  8,  and  whose  product  is  240.  What  numbers 
are  they  1  Ans.   12  and  20. 

4.  The  sum  of  two  numbers  is  =  a,  their  product  =  b. 
What  numbers  are  they  ? 

Ans.  fl  +  v^("-'  — 4ft)   a  —  V{a'  —  U)^ 


5.  It  is  required  to  find  a  number  such,  that  if  we  multi 
ply  its  third  part  by  its  fourth,  and  to  the  product  add  5 


QUADRATIC    EQLATIONS.  109 

times  the  number  required,  the  sum  exceeds  the  number  200 
by  as  much  as  the  number  sought  is  less  than  280. 

Ans.  48. 

6.  A  person  being  asked  his  age,  ansAvcred,  "  My  mother 
was  20  years  old  when  I  was  born,  and  her  age  multiplied 
by  mine,  exceeds  our  united  ages  by  2500."  What  was 
his  age  ?  ^  Ans.  42. 

7.  Determine  the  fortunes  of  three  persons,  A,  B,  C,  from 
the  following  data  :  For  every  $5  which  A  possesses,  B 
has  $9,  and  C  $10.  Farther,  if  we  multiply  A's  money 
(expressed  in  dollars,  and  considered  merely  as  a  num- 
ber) by  B's,  and  B's  money  by  C's,  and  add  both  product? 
to  the  united  fortunes  of  all  three,  we  shall  get  8832. 
How  much  had  each  ? 

Ans.  A  $40,  B  $72,  C  $80. 

8.  A  person  buys  some  pieces  of  cloth,  at  equal  prices, 
for  $60.  Had  he  got  three  more  pieces  for  the  same  sum, 
each  piece  would  have  cost  him  $1  less.  How  many  pieces 
did  he  buy  1  Ans.   12. 

9.  Two  travellers,  A  and  B,  set  out  at  the  same  time, 
from  two  different  places,  C  and  D  ;  A,  from  C  to  D  ;  and 
B,  from  D  to  C.  On  the  way  they  met,  and  it  then  appears 
that  A  had  already  gone  30  miles  more  than  B,  and,  accord- 
ing to  the  rate  at  which  they  travel,  A  calculates  that  he 
q^n  reach  the  place  D  in  4  days,  and  that  B  can  arrive  at 
the  place  C  in  9  days.  What  is  the  distance  between  C 
and  D  ?  Ans.   150  miles. 


fifth  powers  17050.     What  are  the  numbers  ? 

Ans.  3  and  7. 


200 


QUADRATIC    EQUATIONS. 


11.  The  sum  of  two  numbers  is  47,  and  their  product 
546.     Required  the  sum  of  their  squares. 

Ans.   1117. 

12.  The  sum  of  two  numbers  is  20,  and  their  product 
99.     Required  the  sum  of  their  cubes. 

Ans.  2060. 

13.  Divide  the  number  a  into  two  such  parts,  that  the 
sum  of  their  reciprocals  may  equal  b.     What  are  the  parts  1 


Ans 


9 
14.  Divide  -  into  two  such  parts,  that  the  sum  of  their 
2  ^       ' 

reciprocals  may  equal  1.     What  are  the  parts  1 

Ans.  3  and  -. 


15.  Given  the  sum  of  the  squares  of  two  numbers  =a. 
and  the  sum  of  their  reciprocals  =  6;  to  determine  the  num- 
bers. 

Sum  of  numbers  =  TaZ'-  +  2  ±  2(a62_|_  i)n ', 


Ans 


Difference 


lb 


■  2^2{ah-  +  l) 


J 


16.  Find  the  values  of  x  from  the  equation 
3x4-25  7 


21  +  2X 


7  + a:. 


Ans.  X  =  —  7,  or  —  6  i . 


QUADRATIC  EQUATIONS.  201 

17.  Find  the  values  of  x  from  the  equation 

13      ^         4 

Ans.  x=:2  rb^ — 1. 

18.  A  and  B  can  together  perform  a  piece  of  work  in 
two  days,  and  it  wouhl  take  A,  alone,  three  days  longer  to 
perform  it  than  it  would  B  alone.  In  what  time  can  A  and 
B  respectively  perforin  it  1 

.        ^  A  would  require  6  days. 
'■<B      "         "      3     " 

19.  A,  B,  an^  C  agree  to  contribute  $730  towards  build- 
ing a  school-house,  which  is  to  be  at  the  distance  of  2  miles 
from  A,  and  f  of  a  mile  further  from  C  than  from  B.  They 
agree  that  their  shares  shall  be  reciprocally  proportional  to 
their  distances  from  the  school-house.  When  it  was  found 
that  A  paid  $98  more  than  B  paid.  What  was  B's  distance 
from  the  school-house  ?  Ans.  2}  miles. 

20.  I  have  a  certain  number  in  my  thoughts  ;  this  I  mul- 
tiply by  2 J,  add  7  to  the  product,  multiply  this  sum  by  8 
times  the  number  ;  I  then  divide  by  14,  and  from  the  quo- 
tient subtract  four  times  the  number,  and  thus  obtain  2352. 
What  number  is  it  1  Ans.  42. 

21.  Find  two  numbers  such,  that  their  sum  and  product 

together  may  be  =  34,  and  the  sum  of  their  squares  exceed 

the  sum  of  the  numbers  themselves  by  42.     What  are  the 

numbers  1 

4  and  6  ;  or, 
Ans.  ' 


i(— 11  -l-v/— 59),  i(_ll_v/_59). 

22.  It  is  required  to  find  a  number,  consisting  of  three 
digits,  such,  that  the  sum  of  the  squares  of  the  digits,  with- 
out considering  their  position,  may  be  =  104  j   but  the 
26 


202 


QUADRATIC    EQUATIONS. 


square  of  the  middle  digit  exceeds  twice  the  product  of  the 
other  two  by  4  ;  further,  if  594  be  subtracted  from  the  num- 
ber sought,  the  three  digits  become  inverted.  What  num- 
Der  is  it  ?  Ans.  862. 

23.  Find  two  numbers  such,  that  their  sum,  their  pro- 
duct, and  the  difference  of  their  squares  may  be  equal. 

Ans.  ^±W5',^,±W5. 

24.  What  two  numbers  are  they,  whose  sum  is  3,  and 
the  sum  of  whose  fourth  powers  is  17  1 

(  2  and  1 ;  or, 

Ans.  \  , , 

;  J(3  4-n/— 55),  and  J(3— v^— 55). 

25.  What  two  numbers  are  they,  whose  product  is  3,  and 
the  sum  of  whose  fourth  powers  is  82  ? 

(  zir  1,  and  dr  3  ;  or, 

Ans.  \         , , 

(  =b^— 1,  and  =F^— 9. 

26.  A  and  B  can  together  perform  a  piece  of  work  in  3 
days,  and  it  would  take  B  alone  to  do  it  8  days  longer  than 
it  would  take  A.  How  many  days  would  A  alone  require 
to  perform  it  ?  Ans.  4  days. 


Properties  of  the  roots  of  quadratic  equations. 

(154.)  We  have  seen  that  all  quadratic  equations  tun  be 
reduced  to  this  general  form. 

x'  -f  flx  =  5.  (1) 

This,  when  solved,  gives 


Therefore  the  two  values  of  x  are 


(2) 


QUADRATIC  EQUATIONS.  203 


|+\/j^6.  (3) 


(155.)  Now,  since  -={  -  |    is  always  positive   for  all 

real  values  of  a,  it  follows  that  the  sign  of  the  expression 

\-  b,  depends  upon  the  value  of  b. 

(156.)  When  b  is  positive,  or  when  b  is  negative  and  less  than 

— ,  then  will [-  6  be  positive,  and  consequently  y   7+^ 

will  be  real. 

(157.)  When  b  is  negative,  and  numerically  greater  than 

— ,  then \-b  will  be  negative,  and  consequently  V  T  H~  ^ 

will  be  imaginary. 

CASE   I. 


When  Y \-b  in  real. 


1.  If  rt  is  positive,  and  -  is  numerically  greater  than 


Y  ^ — |-  6,  then  will  both  values  of  x  be, r.cfil  and  negative 
2.  When  a  is  either  positive  or  negative,  and  -  is  nume 


rically  less  than  y^-  -f  &,  then  will  both  values  of  x  hi 
real,  the  one  positive  and  th«  other  negative. 


204  QUADRATIC  EQUATIONS. 

3.  When  a  is  negative  and  -  is  numerically  greater  than 
y  — -j-  i,  then  both  values  of  a;  will  be  real  Q.m\  positive. 

CASE    II. 


n/";+- 


When  \/   — \-  a  IS  imaginary. 

In  this  case  both  values  of  x  are  imaginary  for  all  values 
of  a. 

(158.)  When  b  is  negative,  and  numerically  equal  to 

— ,  then  both  values  of  x  become  =  —  -. 

(159.)    If  we   add   together  the  two   values  of  .t,  we 
have 


(-iWf+^)^-(-i-^/^^)=-<.. 

If  we  multiply  them,  we  find 

From  which  we  see, 

That  the  sum  of  the  roots  of  the  quadratic  equation  a:-+ax 
=  h  is  equal  to  —  a. 

And  the  product  of  the  roots  is  equal  to  —  h. 

Hence  the  roots  of  the  equation. 

x^  —  (n  +  ^2)a:  =  — Tiro, 
are  ri  and  r^. 

(160.)  We  can  also  deduce  these  properties  as  follows  • 

If,  in  the  equation  x'^-\-ax=z  6,  we  suppose  the  two  roots 
of  a;  to  be  r^  and  rs,  we  shall  have 


QUADRATIC    EQUATIONS.  205 

ri+ar,  =  b.  (1) 

rl-^ar^=b.  (2) 

Subtracting  (1)  from  (2),  we  find 

,-_,-4_a(ro-;-0  =  0.  (3) 

Dividing  (3)  by  /••,.  —  ?-i,  it  bi^comes 

r.  +  n  +  a  =  0;  (4) 

.  •  •   ri-\-ri  =  —  a.  (5) 

Multiplying  (4)  by  ri,  we  get 

r;ri  +  ?••■;  -f  ar,  =  (\  (6) 

Subtracting  (1)  from  (6),  we  get 

r,r,  =  — /j.  (7) 

Equations  (5)  and  (7)  correspond  with  the  properties  just 
found,  Art.  159. 

(161.)  We  have  seen  that  every  quadratic  equation,  when 
solved,  gives  two  values  for  the  unknown  quantity.  These 
values  will  both  satisfy  the  algebraic  conditions,  and  some- 
times they  will  both  satisfy  the  particular  conditions  of  the 
problem,  but  in  most  cases  but  one  value  of  the  unknown  is 
applicable  to  the  problem  ;  and  the  value  to  be  used  must 
be  determined  from  the  nature  ol  the  question. 

We  will  illustrate  this  principle  by  the  solution  of  some 
particular  questions, 

1 .  Find  a  number  such  that  its  square  being  subtracted 
from  five  times  the  number,  shall  give  6  for  remainder. 
Let  x=  the  number  sought. 
Then,  by  the  conditions  of  the  questions,  we  have 

5x— .r*=6.  (1) 

Changing  all  the  signs  of  (1),  it  becomes 

j-3_5:c3=  — 6.  (2) 

which,  wdien  solved  by  the  rule  for  quadratii-s,  gives 


206 


QUADRATIC    EQUATIONS. 


5±1 


3,  or  2. 


Taking  the  first  value  of  x  =  3,  we  find  its  square  to  be  9. 

Five  times  this  value  of  a;,  is  5  X  3  =  15. 

And  15  —  9  =  6;  therefore  the  number  3  satisfies  th' 
question. 

The  number  2  will  satisfy  it  equally  well,  since  its  squan 
=  4,  which,  subtracted  from  five  times  2  =  10,  gives  fo 
remainder  6. 

2.  Find  a  number  such  that  when  added  to  6,  and  th. 
sum  multiplied  by  the  number,  the  product  will  equal  th« 
number  diminished  by  6. 

Let  a;  =  the  number  sought ;  then,  by  the  conditions  of 
the  question,  we  have 

{x-\-6)x=x  —  6.    '  (1) 

Expanding  and  collecting  terms,  we  find 

x-^-\-5x=  —  6.  (2) 

This  solved  gives 

-5±1 


Here,  as  in  the  last  question,  we  find  that  both  values  of 
X  will  satisfy  our  question. 

If  we  take  the  first  value,  x  =  —  3,  we  find  that  the  num 
ber  —  3  added  to  6  gives  3,  which  multiplied  by  —  3  give? 
—  9  ;  and  this  is  the  same  as  —  3  diminished  by  6. 

If  we  take  the  second  value,  x  =  —  2,  we  find  that  the 
number  — 2  added  to  6  gives  4,  which  multiplied  by  —  2 
gives  — 8  ;  and  this  is  the  same  as  —  2  diminished  by  6. 

3.  Find  a  number  which  subtracted  from  its  square,  shall 
give  6  for  remainder. 

Let  x=  the  number,  then  we  have 


QUADRATIC    EQUATIONS. 

207 

X'  —  X:=e. 

(1) 

x-^^^-3,or_2. 

This  gives 


If  we  take  3  for  the  number,  its  square  is  9,  from  which 
subtracting  3,  we  have  6. 

Again,  taking  — 2  for  the  number,  its  square  is  4,  from 
which  subtracting  — 2,  we  have  6. 

So  that  both  values  of  x  satisfy  the  conditions  of  the  ques- 
tion. 

4.  A  and  B  travel  from  the  same  place,  and  in  the  same 
direction.  The  first  day  A  travels  but  1  mile,  the  second 
day  he  goes  3  miles,  the  third  day  5  miles,  and  so  on  in 
arithmetical  progression.  After  A  has  been  gone  8  days, 
B  follows,  travelling  uniformly  at  the  rate  of  30  miles  each 
day.  How  many  days  after  B  starts  will  they  be  to- 
gether ? 

Let  X  =  the  number  of  days  sought. 
Then  will  z-f-8=  the  number  of  days  which  A  travelled, 
(a:4-8)  =  distance  travelled  by  A. 
36a:  =       ••  "  B. 

Hence, 

{x  -I-  Sy-  =  360-. 

This,  solved  by  the  usual  method  of  quadratics,  gives 
j-=:4,  or  X  =:  16. 

From  which  we  learn  that  they  were  twice  together.  First 
B  overtakes  A  at  the  end  of  4  days,  and  then  in  12  days 
more  A  overtakes  B. 

In  this  case  both  answers  arc  aj^plicable. 

.   By  selling  a  watch  for  $24,  I  lose  as  much  per  cent 
as  tiie  watch  cost  me.     What  was  the  cost  of  the  watch  '? 


208 


QUADRATIC    EQUATIONS. 


Let  X  =  the  number  of  dollars  the  watch  cost. 
Then  will  x  —  24  =  the  loss  incurred  by  the  sale. 
The  loss  per  cent,  will  be 

(x  — 24)100 


Therefore,  by  the  question,  we  have  this  equation  : 
100(a;  — 24)_ 

X 

This  gives  x  =  40,  or  a:  =  60. 

In  this  case,  also,  both  answers  are  applicable. 

6.  There  is  a  number  consisting  of  two  digits,  of  which 
the  right-hand  digit  is  3  greater  than  the  left-hand  digit, 
and  the  number  itself  is  equal  to  the  square  of  the  right- 
hand  digit.     What  is  the  number  ? 

Ans.  25,  or  36. 

7.  A  number  consists  of  two  digits,  of  which  the  right- 
hand  digit  is  double  the  left-hand  digit.  The  number  ex- 
ceeds the  square  of  the  right-hand  digit  by  8.  What  is  the 
number  1  Ans.   12,  or  24. 

8.  A  and  B  speaking  of  their  ages,  A  said  he  was  15 
years  older  than  B,  and  that  the  square  of  his  age  was  equal 
to  64  times  B's  age.     What  were  their  ages  ? 

A  =--  40  and  B  =:  25  ;  or. 
A  =  24  and  B  =-.    9. 


Ans. 


9.  By  the  law  of  universal  attraction  we  knoic,  that  the 
attraction  of  different  bodies,  at  different  distances,  varies  di- 
rectly as  their  masses  and  inversely  as  the  squares  of  theii 
distances  from  the  attracted  point. 

The  above  law  being  admitted,  it  is  required  to  find  a 
point  in  the  right  line  which  joins  the  centres  of  the  two 
spherical  bodies,  whose  masses  arewiiand 


QUADRATIC    KQIATIONS.  209 

(ills  point  will  be  attracted  wltli  equal  force  by  each  of  the 
bodies. 

p~  i        -j-~— — ]         ——————— 

Pi  mi      })'  p      1712  pi 

Let  mi  and  7n-2  be  the  position  of  the  bodies. 

Let  the  distance  between  the  centres  of  the  two  bodies  mi 
and  m^y  =  d. 

Also,  let  p  denote  the  point  sought". 

Put  a;---mip  =  the  distance  from  the  body  wii  to  the 
point  sought,  measured  from  mi  towards  the  right. 

Then  d  —  x  =  m-ip  =  the  distance  from  the  body  mo  to. 
the  point,  measured  from  mo  tow^ards  the  left. 

Now,  having  reference  to  the  above  law,  we  know  that 
the  attractions  of  the  two  bodies  upon  the  point  p  will  be  to 
each  other  as  the  expressions 

mi  ma 

But,  by  the  questions,  these  forces  of  attraction  are  equal ; 
therefore  we  have' this  condition  : 
rni_      mo 

x^  —  {d-xY'  ^ 

Extracting  the  square  root  of  both  members  of  (1),  we 
have 

y/mi  _  db  y/ma  .    . 

X  d—x  '  ^    ' 


This  reduced,  by  rules  for  simple  equations,  gives 
Hence, 


mip  =  x  = ^1 — ■ Xd.  (3) 

v/wii  ±  s/ma 


m^p  =  d  —  x= xd.  (4) 

y/mi  zfc  v/ma 

27 


210  QUADRATIC  EQUATIONS. 

If  we  use  the  upper  signs,  we  get 


niip 


y/lTii  -\-  y/mo 


(A) 


m^ip  = ^--- X  d. 

y/nii  -\-  y/m-z 


By  taking  the  lower  signs,  we  have 


mxp 


y/nii  —  \/m.i2 


'■} 


///o»  = X  d. 


(B) 


We  will  now  interpret  these  expressions  for  different  nu- 
merical values  of  c/,  wii,  mo,  and,  in  order  that  the  following 
reasoning  may  be  rigidly  correct,  it  is  necessary  to  suppose 
all  the  matter  of  the  bodies  W]  and  mo  to  be  concentrated 
at  the  centres  of  the  bodies. 

CASE   I. 

When  d  =  a  finite  quantity. 
And  mi  >  ma. 

In  this  case,  we  evidently  have 


v/mi  +  v/mo 
N/m2 


>i  and  <1 


Consequently,  the  first  set  of  values,  denoted  by  (A),  give 
m\p  =  a  positive  quantity  which  is  <d,  but  >-. 

imp  =  a  positive  quantity  which  is  <-. 


QUADRATIC    EQUATIONS.  21] 

These  values  give  for  the  point  sought,  a  position  between 
wii  and  7712,  but  nearer  wio  than  mi. 
Again, 


a  negative  quantity. 


v////l y/m-2 

Therefore,  the  second  set  of  values,  denoted  by  (B),  give 
mijj  =  a  positive  quantity  which  is  ></. 
jn'2p  =  a  negative  quantity. 
Now,  since  the  distances  from  wo,  measured  towards  the 
left,  ar£  consit'iered  as  positive,  the  distances  in  an  opposite 
'.lirection  must  be  regarded  as  negative. 

Plence,  these  second  values  give  for  the  point  a  position 
on  the  right  of  ??io. 

CASE   11. 

W'ie?i  d  =  a  finite  quantity, 
.ind  THi'C.rni. 

In  this  case 

v/Wli  , 

~, — ; — : — ^2' 

>i  and  <1. 


Consequently  the  first  set  of  values,  denoted  by  (A),  give 

.        d 

mfp  =  a  positive  quantity  >-. 

.     ^d 
map  =z  a  positive  quantity  >-. 

And  the  point  lies  between  mi  and  m-j,  nearer  7ni  than  Wj. 


212 


QUADRATIC    EQUATIONS. 


Vmi 


v/mi  —  s/ni-2 


a  negative  quantity, 
a  positii-e  quantity  >  1. 


Tlierefore  these  second  values  give  for  the  point  a  position 
on  the  left  of  wii. 

This  case  is  obviously  the  same  as  Case  I.,  when  we  in- 
terchange the  bodies  mi  and  wia. 


CASE   III, 


When  d  =  a  finite  quantity. 
And  mi  =  m^. 


In  this  case, 


v/mi 


■v/Wii-j-x/'mg 


Consequently,  the  first  set  of  values,  denoted  by  (A),  give 

d 
mip  =  -. 


And  the  point  is  equi-distant  from  nii  and  7nQ. 
Again, 


■v/mi 


-v/mi 


-=dr !z=±an  infinite  quantity,  (Art. 133. 

=^  -      '=^an  infinite  quantity.  (Art.  133, 

Vwii  —  -/wa  0 


QUADRATIC    EQUATIONS.  213 

Therefore,  the  second  set  of  values,  denoted  by  (B),  give 
for  the  point  a  position  at  an  infinite  distance  either  to  the 
right  or  left. 

CASE    IV. 

Wien  d  =0.  ^nd  mi\.  771-2. 

In  this  case,  we  have 

imp  =  0, 
m-2p  =  0, 

for  both  sets  of  values  ;  consequently  there  is  but  one  point 
which  is  equally  attracted  by  both  bodies,  and  that  point  is 
the  comiron  centre  of  the  two  bodies. 


CASE    V. 

W/ien  d^O.  Jind  mi  =  mo. 

The  first  set  of  values  evidently  become 

imp  =  0, 

m-ip  =  0. 

Which  shows  that  the  point  is  in  the  common  centre  of  the 
two  bodies. 

The  second  set  of  values  give 
7ni;p  = -=- an  indeterminate  quantity.  (Art.  134.) 

TTi^p  =  -  =  an  indeterminate  quantity.   (Art.  134.) 

So  that  the  point  may  be  any  where  on  the  line  which  join.s 
the  centres  of  the  bodies.  Since  the  two  centres  are  united, 
every  line  which  passes  through  this  common  point  may  be 
regarded  as  joining  those  centres  j  consequently  every  point 


214  QIADUATIC    EQUATIONS. 

in  space  is,  in  this  particular  case,  equally  attracted  by  each 
body. 

From  the  above  discussion,  we  see  that  the  analytical  ex- 
pressions are  faithful  to  give  all  the  particular  cas^s  which 
are  possible  to  arise  from  giving  particular  values  to  the 
constant  quantities  which  enter  into  the  conditions  of  the 
question. 

(162.)  We  will  now  add  a  couple  examples  for  the  pur- 
pose of  illustrating  the  case  in  which  the  roots  are  imagi- 
nary. 

1.  Find  two  numbers  whose  sum  is  8,  and  whose  pro- 
duct is  17. 

Let  x  =  one  of  the  numbers,  then  will  8 — x=  the 
other  number. 

The  product  is  (8  —  x)x  --=8x  —  x"^,  which,  by  the  con- 
ditions of  the  question,  is  17. 

Therefore,  we  have  this  equation  of  condition, 

xi  —  Sx=—ll.  (1) 

This,  solved  by  the  usual  rules  for  quadratics,  gives 
x  =  4  db  "^ —  1 5  for  one  of  the  numbers, 
and  8  —  (4  ±  V—l)  =  (4  =F  v/_l,)  for  the  other  num- 
ber. 

(  4  _j_\/] I 

Therefore,  the  numbers  are  <  I 

(4=F^/— 1, 

both  of  which  are  imaginary  ;  we  are  therefore  authorized 
to  conclude  that  it  is  impossible  to  find  two  numbers  whose 
sum  is  8,  and  proihict  17. 

We  may  also  satisfy  ourselves  of  this  as  follows  :  Since 
the  sum  of  the  two  numbers  is  8,  they  must  average  just  4  j 
h'.'uce  the  greater  must  exceed  4  just  as  much  as  the  less 


QUADRATIC    EQUATIONS.  215 

falls  short  of  4.     Therefore  any  t\YO  numbers  whose  sum  !•? 
8  may  be  represented  by 

4  — a-. 
Taking  their  product,  we  have 

{4-\-x){4:  —  x)=ie  —  x. 
Now,  since  x-  is  positive  for  all  real  values  of  x,  it  fol- 
lows that  the  product  16  —  x-  is  always  less  than  16  ;  that 
is,  no  two  real  nu77ibers  whose  sum  is  8,  can  be  found  such 
that  their  product  can  equal  17. 

If  we  put  the  expression  for  the  product,  which  we  have 
just  found  equal  to  17,  we  shall  have 
16  —  x-  =  17, 

consequently,      x  =  zb  ^ —  1. 

And,4  +  a:  =  4zb^/^,)  ^  ,  r       ,  u     i, 

. >  the  same  values  as  lound  by  the 

4_a;  =  4zpv/_i,^ 

first  method. 

These  values,  although  they  are  imaginary,  will  satisfy 
the  algebraic  conditions  of  the  question  ;  that  is,  their  sum  is 

(4±N/iri)+(4:^v/Zn:)  =  8, 
and  their  product  is 

(4  ±  ^/=^l)  X  (4  T  v/— 1)  =z  17. 

2.  Find  two  numbers  whose  sum  is  2,  and  sum  of  their 
reciprocals  1. 

Denoting  the  numbers  by  x  and  y,  we  have  the  following 
relations  : 


These,  solved  by  the  ordinary  rules,  give 


216 


QUADRATIC    EQUATIONS. 


(2) 


x  =  1±n/— 1, 

Both  lirj:\  values  are  imaginar}  ;  consequently  the  condi- 
tions of  the  question  are  absurd. 

We  may  also  show  the  impossibility  of  this  question  as 
follows  :  The  sum  being  2  the  numbers  may  be  denoted  by 

l  +  o:,) 

1  — a:.  > 
Taking  the  sum  of  their  reciprocals,  we  have 

1+x^l— x' 
which,  when  reduced  to  a  common  denominator,  becomes 
2 


1  — x2 

The  denominator  of  this  expression  cannot  be  greater 
than  1 ;  for  all  real  values  of  x,  the  expression  must  ex- 
ceed 2.  Therefore,  it  is  imfossihle  to  find  two  numbers 
whose  sum  shall  equal  2,  and  sum  of  their  reciprocals  equal  1. 

(163.)  From  what  has  been  said,  we  conclude  that  when, 
in  the  course  of  the  solution  of  an  algebraic  problem,  we 
fall  upon  imaginary  quantities,  there  must  be  conditions  in 
the  problem  which  are  incompatible. 

Under  Art.  128,  we  remarked  that  imaginary  quantities 
had  been  advantageously  employed  as  aids  in  the  solution 
of  many  refined  and  delicate  problems  of  the  higher  parts 
of  analysis  ;  here  we  notice  their  utility  in  pointing  out  the 
impossibility  of  questions,  which  otherwise,  with  only  a  su 
perficial  investigation,  might  be  supposed  possible. 


ARITHMETICAL    PROGRESSION.  21' 


CHAPTER  VI. 


flATIO  AND  PROGRESSION. 

(164.)  By  Ratio  of  two  quantities  we  mean  their  relation. 
When  we  compare  quantities,  by  seeing  how  much  greater 
one  is  than  another,  we  obtain  arithmetical  ratio.  Thus  : 
the  arithmetical  ratio  of  6  to  4  is  2,  since  6  exceeds 4  by  2  ; 
in  the  same  way,  the  arithmetical  ratio  of  11  to  7  is  4. 

In  the  relation         a  —  c=ir.,  (1) 

r  is  the  arithmetical  ratio  of  a  to  c. 

The  first  of  the  two  terms  which  are  compared  is  called 
the  antecedent;  the  second  is  called  the  consequent.  Thus, 
referring  to  (1),  we  have 

a  =  antecedent. 
c  =  consequent, 
r  =  ratio. 
From  (1),  we  get  by  transposition, 

«  =  c  +  r,  (2) 

c  =  a  —  r.  (3) 

Equation  (2)  shows,  that  in  an  arithmetical  ratio  the  an 
tecedent  is  equal  to  the  consequent  increased  by  the  ratio. 

Equation  (3'  in  like  manner  shows,  that  the  consequent 
is  equal  to  the  antecedent  diminished  by  the  ratio. 
28 


218  ARITHMETICAL    PROGRESSION. 

(165.)  When  the  arithmetical  ratio  of  r,ny  two  terms  is 
the  same  as  the  ratio  of  any  other  two  terms,  the  four  terms 
together  form  an  aiithmetical  proportion. 

Thus,  if  a  —  c  =  r  ;  and  a' —  c'=  r,  then  will 

a — c  =  a' — c',  (4) 

which  relation  is  an  arithmetical  proportion,  and  is  read 
thus  : 

a  is  as  much  greater  than  c,  as  a'  is  greater  than  c'. 
Of  the  (our  quantities  constituting  an  arithmetical  pro- 
portion, the  first  and  fourth  are  called  the  extremes,  the 
second  and  third  are  called  the  means. 

The  first  and  second,  together,  constitute  the  first  coup- 
let;  the  thiril  and  fourth  constitute  the  second  couplet. 
From  equation  (4),  we  get  by  transposing, 

a  +  c'=  a'-j-  c,  (5) 

which  shows,  that  the  sum  of  the  extremes,  of  an  arithmeti 
cal  proportion,  is  equal  to  the  su)n  of  the  means. 
If  c  =  a',  then  (4)  becomes 

a  —  a'=a' — c',  (6) 

which  changes  (5)  into 

a-\-c'=2a'.  (7) 

So  that,  if  three  terms  constitute  an  arithmetical  pro 
portion,  the  sum  of  the  extremes  will  equal  twice  the  mean. 

(166.)  A  series  of  ([r.antities  which  increase  or  decrease 
by  a  constant  difference  form  an  aiithmetical  progression. 
When  the  series  is  increasing,  it  is  called  an  ascending  jyro- 
gression;  when  decreasing  it  is  called  a  descending  progres- 
sion. 

Thus,  of  the  two  series 

1,  3,  5,  7,  9,  11,  &c.  (8) 

27,  23,  19,  15,  11,  7,  &c.  (9) 


ARITHMETICAL    PROGRESSION.  219 

The  first  is  an  ascending  progression,  whose  ratio  or  com- 
mon  difference  is  2  ;  the  second  is  a  descending  progression, 
whose  common  difference  is  4. 

(167.)  If  a  =  the  first  term  of  an  ascending  arithmeti- 
cal progression,  whose  common  difference  =  J,  the  succes- 
sive terms  will  be 

fl  =  first  term, 
a-\-  d=  second  term, 
a-\-2d  =  third  term, 
a  +  3d=  fourth  term,  V  (\Q\ 


a  -\-  {n  —  l)d  =  Tith  term. 
If  we  denote  the  last  or  nth  term  by  /,  we  shall  have 

l  =  a-{.{n—\)d.  (11) 

From  (11)  we  readily  deduce 

a  =  l  —  in—\)d,  (12) 

"  =  '-=^+1.  (14) 

When  the  progression  is  descending,  we  must  write  —  d 
for  d  in  the  above  formulas. 

Suppose,  in  an  arithmetical  progression,  a;  to  be  a  term 
which  is  preceded  by  q  terms  ;  and  y  to  be  a  term  which  is 
followed  by  q  terms  ;  then  by  using  (11)  we  have 

x^a-\-qd^  (15) 

y  =  l—qd.  (16) 

Taking  the  sum  of  (15)  and  (16),  we  get 

x  +  y  =  a-\-L  (17) 


220  ARITHMETICAL    PROGRESSION. 

That  isj  the  sum  of  any  two  terms  equi-distant  from  the 
extremes  is  equal  to  the  sum  of  the  extremes,  so  that  the 
terms  will  average  half  the  sum  of  the  extremes ;  conse- 
quently, the  sum  of  all  the,  terms  equals  half  the  sum  of  the 
extremes  multiplied  by  the  number  of  term^. 

Representing  the  sum  of  n  terms  by  s,  we  have 

s  =  -l-xn.  (18) 

From  (18)  we  easily  obtain 


;  =  -—/.  (19) 


n 


l=-~a.  (20) 


(21) 


2s 

Any  three  of  the  quantities 

a  =  the  first  term, 

d  =  common  difference, 

n  =  number  of  terms, 

/  =  last  term, 

s  =  sum  of  all  the  terms, 

being  given,  the  remaining  two  can  be  found,  which  must 
give  rise  to  20  different  formulas,  as  given  in  the  following 
table  for  Arithmetical  Progression. 

(168.)  We  have  not  deemed  it  necessary  to  exhibit  the 
particular  process  of  finding  each  distinct  formula  of  the  fol- 
lowing table,  since  they  are  all  derived  from  the  two  fun- 
damental ones,  (1)  and  (7) ,  by  the  usual  operations  upon 
equations  not  exceeding  the  second  degree.  It  will  furnish 
a  good  exercise  for  the  student  to  deduce  all  these  formulr.s 
by  the  aid,  only,  of  formulas  1  and  7. 


ARITHMETICAL    PROGUESSIOX. 


2-2\ 


'    No. 

Given. 

Requi- 
red. 

Formulas. 

Corr.  , 

17 
19 

20 

18 

1 

2 
3 

4 

a,  dj  n 
a,  rf,   s 

a,  n,  5 

/ 

l  =  a-\-{n  —  l)d 

/  =  — Jd±v/2d*+(a  — i(i)- 

I— -  —  a 
n 

s       {n-l)d 

n~^        2 

5 
6 

!    7 
8 

a,   d,  / 
a,  ?j,   / 

S 

s=zln[2a-\'i^ji—l)d] 

'          2       '              2d 

s  =  hi{a+l) 

s  =  i7i\2l  —  {n^l)d\ 

8 
5 

1    9 

!io 
11 

'l2 

a,  n,  5 
a,   /,   5 
n,   /,   s 

d 

71  —  1 

^       25  —  2an 
Hn-1) 

2,.  _  /  _  a 
2nl—2s 
n(7i-l) 

12 
10 
16 
14 

113 
14 

i  15 
16 

a,  d,   I 
0,    d,   s 
a,   /,  V 
d,    /,   5 

n 

d  —  2a      J 2s  ,   /2a  — dV- 
2s 

2l  +  d  ,  ,  //2/+d\2     2. 
^^-    2d    ^V(   2d   )-d 

17 
18 

19 
20 

= 

d,  n,  / 

d,   n,  5 

r/,    /,   s 
n,   /,   i 

a 

a=/  — (n— l)d 
5        (n— l)d 
"-n             2 

1 
4 

2 
3 

a  =  idd=N/(/-f-^dr  — 2d^ 

2^         ; 

n 

222  ARITHMETICAL    PROGRESSION. 

(169.)  From  the  nature  of  an  arithmetical  progression, 
we  discover  that  if  we  subtract  the  common  difference  from 
the  last  term,  we  shall  obtain  the  term  next  to  the  last ;  if 
we  subtract  from  the  last  term  twice  the  common  difference, 
we  obtain  the  second  term  from  the  last.  Hence  the  terms 
of  an  arithmetical  progression  will  be  reversed  if  we  inter- 
change the  values  of  a  and  /,  and  at  the  same  time  change 
the  sign  of  d.  Thus,  the  general  form  of  an  arithmetical  pro* 
gression  is 

0,  ct+rf,  a+2rf, /  — 2rf,  /  — (/,  I. 

Changing  a  to  /,  /  to  «,  and  changing  the  sign  of  d^  we 
have 

l^  I  —  d)  /  —  2(/, a+St/,  a-\-d^  a, 

which  is  precisely  the  same  progression  as  the  first,  with 
the  terms  arranged  in  a  reverse  order.  The  above  change 
has,  of  course,  no  effect  upon  the  number  of  terms,  nor  upon 
the  sum  of  all  the  terms. 

Therefore,  in  any  of  the  formulas  of  the  preceding  table 
we  are  at  liberty  to  make  the  above  named  changes.     As  an 
example,  we  will  take  from  the  table  formula  2,  which  is 
/  —  —  Jrf  =b  ^2ds-{-{a—ldf. 

Now,  changing  /  to  a,  a  to  /,  and  changing  the  sign  of  d. 
it  becomes 

u  .=  Id  ±  V(/+;(i)s  — 2rf^, 
which  is  formula  19. 

In  the  same  way,  formulas  14  and  16  may  be  deduced 
from  each  other.  Such  formulas  as  may  be  derived  from 
each  other  by  the  above  changes  we  shall  call  correlative 
formulas.  It  is  evident  that  some  of  the  formulas  of  the  ta- 
ble have  no  correlative.  Thus,  formulas  13  and  15  are  not 
altered  by  the  above  changes.  Those  formulas  which  have 
correlative  formulas  have  them  referred  to  in  the  table,  un- 
der column  headed  Corr. 


ARITHMETICAL    PROGRESSION.  223 


1.  The  first  term  of  an  arithmetical  progression  is  7,  the 
common  difference  is  |,  and  the  number  of  terms  is  16. 
What  is  the  last  term  ? 

To  solve  this,  we  take  formula  1  from  our  table,  which  is 
l  =  a-ir{n  —  l)d. 
Substituting  the  above  given  values  for  a,  d,  and  7i,  we  find 
;  =  7  +  Kl6— l)=10f. 

2.  The  first  term  of  an  arithmetical  progression  is  |,  the 
common  difference  is  |,  and  the  last  term  is  3|.  What  is 
the  number  of  terms  1 

In4.his  example  w-e  take  formula  13. 
I  — a 

which  in  this  present  case  becomes 

n  =  ^i^-|-l  =26. 

3.  One  hundred  stones  being  placed  on  the  ground  in  a 
straight  line,  at  the  distance  of  two  yards  from  each  other, 
how  far  will  a  person  travel  who  shall  bring  them  one  by  one 
to  a  basket,  placed  at  two  yards  from  the  first  stone  '? 

In  this  example  a— -4;  d=4:  ;  7i  =  ]00;  which  values 
being  substituted  in  formula  5,  give 

6>  =z  50  j  8  -f  99  X  4  I  =  20200  yards, 

which,  divided  by  1760,  the  number  of  yards  in  one  mile, 
we  get 

s  =  11  miles,  840  yards. 

4.  What  is  the  sura  of  n  terms  of  the  progression 

1,3,5,7,9,    ? 

Ans.  .s-  =  «'■'. 


524  GEOMETRICAL    PROGRESSION. 

5.  What  is  the  sum  of  ti  terms  of  the  progression 
1>2,3,4,5,   ■? 

A„s.  .^±±n. 


GEOMETRICAL    RATIO. 

(170.)  When  we  compare  quantities  by  seeing  how  many 
times  greater  one  is  than  another,  we  obtain  geometrical 
ratio.  Thus  the  geometrical  ratio  of  8  to  4  is  2,  since  8  is 
2  times  as  great  as  4.  Again,  the  geometrical  ratio  of  15 
to  3  is  5. 

In  the  relation,  -  =r,  (1) 

c 

r  is  the  geometrical  ratio  of  a  to  c. 

As  in  arithmetical  ratio, 

a  =  antecedent^ 

c  =  consequent^ 

r  =  ratio. 

From  (1),  we  get  by  reduction, 

a  —  cr^  (2) 

.=2.  (3) 

Equation  (2)  shows,  that  in  a  geometrical  ratio  the  ante- 
cedent is  eqiial  to  the  consequent  multiplied  by  the  ratio. 

Equation  (3)  shows,  that  the  consequent  is  equal  to  the 
antecedent  divided  by  the  ratio. 

(171.)  When  the  geometrical  ratio  of  any  two  terms  is 
the  same  as  the  ratio  of  any  other  two  terms,  the  four  terms 
together  form  a  geometrical  proportion. 


Thus,  if -=  r;  and  -.  =  r,  then  will 
c  c 


1=?'  w 


GEOMETRICAL    TROGRESSIOX.  225 

which  relation  is  a  geometrical  proportion,  and  is  generally 
Written  thus  : 

a   :  c   :   :  o'   :  c',  (5) 

which  is  read  as  follows  :  a  is  to  c,  as  a'  is  to  c'. 

Of  the  four  quantities  which  constitute  a  geometrical  pro- 
portion, as  in  arithmetical  proportion,  the  first  and  fourth 
are  called  the  extremes,  the  second  and  third  are  called  the 
7neans. 

The  first  and  second  constitute  the^r^^  couplet ;  the  third 
and  fourth  constitute  the  second  couplet. 

From  equation  (5),  or  its  equivalent  (4),  we  find 

ac'  =  a'c,  (6) 

which  shows,  that  the  product  of  the  extremes  of  a  geome- 
trical proportion,  is  equal  to  the  product  of  the  means. 

If  c  =  a',  then  (5)  becomes 

a   :  a'   :   \  a'   :  c',  (7) 

which  changes  (6)  into 


ac  =  a 


(8) 


so  that  if  the  two  means  which  constitute  a  geometrical  pro 
portion  be  equal,  then  the  product  of  the  extremes  will  equal 
the  square  of  the  mean. 

(172.)  Quantities  are  said  to  be  in  proportion  by  inver- 
sion, or  inversely,  when  the  consequents  are  taken  as  ante- 
cedents, and  the  antecedents  as  consequents. 

From  (5),  or  its  equivalent  (4),  which  is 

we  have,  by  inverting  both  terms, 
c  ^c_[ 
a       a'' 
Therefore,  by  Art.  171, 

c  :  a  :   :  c'   :  a'.  (10) 

29 


226  GEOMETRICAL    PROGRESSION. 

W/iich  shows  ^  tJiat  if  four  quantities  are  in  proportion  they 
will  be  in  proportion  by  inversion. 

(173.)  Quantities  are  in  proportion  by  alternation^  or  al- 
ternately., when  the  antecedents  form  one  of  the  couplets, 
and  the  consequents  form  the  other. 

Resuming  (4), 

-=^-  .     (11) 

c       c 

c 
Multiplying  both  terms  of  (11)  by  —,  it  will  become 

o 

a  c 

a'       c' 

Therefore,  by  Art.  171, 

a   :  a'    :   :  c   :  c'.  (12) 

Which  shows,  that  if  four  quantities  are  in  proportion  they 

will  be  so  by  alternation. 

(174.)  Quantities  are  in  proportion  by  co7nposition,\\hen 

the  sum  of  the  antecedent  and  consequent  is  compared  either 

with  antecedent  or  consequent. 

Resuming  (4), 

c     c' 
If  to  (13)  we  add  the  terms  of  the  following  equation  -=:-, 

each  of  whose  members  is  equal  to  unity,  we  have 
a-fc  _a'-\-c' 
c      ~     c'    ' 
Therefore,  by  Art.  171, 

a-\-c  :  c  :   :  a'-\-c'   :  c'.  (14) 

Which  shows,  that  if  four  quantities  are  in  proportion  they 
will  he  so  by  composition. 


GEOMETRICAL   PROGRESSION.  227 

(175.)  Quantities  are  said  to  be  in  proportion  by  division, 
when  the  difference  of  antecedent  and  consequent  is  com- 
pared with  either  antecedent  or  consequent. 

c       c' 
If  we  subtract  the  equation  -  =  -.,    each    member    of 
c      c 

which  is  equal  to  1,  from  equation  (4) ,  we  find 


c  c 

Therefore,  by  Art.  171,  we  have 

a  —  c  :  c  :   :  a' — c'  :  c'.  (15) 

Which  shows,  that  if  four  quantities  are  in  proportion,  they 
vAll  be  so  by  division. 
Equation  (4)  is 

a a' 

c       c'' 
Raising  each  member  to  the  nth  power,  we  have 

0"  _  a'" 

Therefore,  by  Art.  171,  we  have 

a"  :  c"  :   :  a'"  :  c'".  (l6) 

Which  shows,  that  if  foxir  quantities  are  in  proportion,  likt 

powers  or  roots  of  these  quantities  will  also  be  in  proportion. 

If  we  have  a  :  c  :   :  a'     :  c',     ^ 

a  :  c  :   :  a"    :  c".    > 


(17) 


a  '.  c  :   :  a       :  C 


&c.,  &c. 

These  give  by  alternation,  Art.  173, 


&€.,  &c. 


22S 


GEOMETIUCAL    PBOGRESSION, 


Tiierefore,  by  inversion,  Art.  172,  we  have 


We  also  have 


c 
c 

c" 
c 

a  c 

&c.,  &c, 

0 c 

n       c 


(18) 


.(19) 


Taking  the  sum  of  equations  (18),  we  have 

aJ^a'-\.a"J^a">  j^S^c.  _  c  +  c'-\- c" -\- c'" -\- kc. 

a  c 

Therefore,  by  Art.  171,  we  have 
a-\-a'-\-a"-\-a"'-\-kc.:a:   :  c4-c'  +  c"+c"'+&c.:  c.   (20) 

Which  shows^  that  if  any  number  of  quantities  are  propor- 
tional^ the  sum  of  all  the  antecedents  will  he  to  any  one 
antecedent^  as  the  sum  of  all  the  consequents  is  to  its  corres- 
ponding consequent. 
(176.)  If  we  have 


a    :  c    '.      :  a     :  c  , 

a":c":      :a"':c"', 

we  find 

a      a' 

(21) 

a"       a'" 
7'~  c'"' 

(22) 

Multiplying  together  the   equations   (21)   and   (22),  we 
have 

aXa"       a'Xa'" 


cX  c"       c'Xc'" 


(23) 


GEOMETRICAL    PROGRESSION.  229 

Therefore,  by  Art.  171,  we  have 

aXa"  :  cXc"  :   :  a' Xa'"  :  c' Xc'".  (24) 

Which  shows,  that  if  there  be  two  sets  of  proportional  quan- 
tities, the  products  of  the  corresponding  terms  will  he  pro- 
portional. 

(177.)  A  series  of  quantities  which  increase  or  decrease 
by  a  constant  multiplier  forms  a  geometrical  progression. 
When  the  series  is  increasing,  that  is,  when  the  constant 
multiplier  exceeds  a  unit,  it  is  called  an  ascending  progres- 
sion; when  decreasing,  or  when  the  constant  multiplier  is 
less  than  a  unit,  then  it  is  called  a  descending  progression. 

Thus,  of  the  two  series, 

1,  3,  9,  27,  81,  243,  &c.,  (25) 

256,  128,  64,  32,  16,  8,  &c.  (26) 

the  first  is  an  ascending  progression,  whose  constant  multi- 
plier or  ratio  is  3  ;  the  second  is  a  descending  progression, 
whose  ratio  is  ^. 

(178.)  If  a  is  the  first  term  of  a  geometrical  progression, 
whose  ratio  =^  r,  the  successive  terms  w'ill  be 
a=  first  term, 
ar  =  second  term, 
ar-=  third  term, 
aH=  fourth  term,  V  (27) 


nth  term. 


If  we  denote  the  last  or  nth  terra  by  /,  we  shall  have 

l=ar"-K  (28), 

If  we  represent  the  sum  of  n  terms  of  a  geometrical  pro- 
gression by  s,  we  shall  have 
s  =  a-\-ar-^  ar^-f  ar"-^ . . .  .-j-  ar'^'  +  or"-'.  (29) 


230 


GEOMETRICAL    PROGRESSION. 


Multiplying  all  the  terms  of  (29)  by  the  ratio  r,  we  have 
rs  =zar  -{■  ar--\-  ar^-\-  ar'^-^ . . . .  +  ar'^^-\-  ar^.        (30) 
Subtracting  (29)  from  (30),  we  get 

(r— l)s  =  G(r"— 1). 


Therefore, 


==a.  <  > 

Ir-lS 


(31) 
(32) 


Any  three  of  the  quantities 

a  =  first  term, 

r  =  ratio, 

n  =  number  of  terms, 

/=  last  term, 

s  =  sum  of  all  the  terms, 
being  given,  the  remaining  two  can  be  found,  which  as  in 
arithmetical  progression,  must  give  rise  to  20  different  for- 
mulas, as  given  in  the  following  table   for  Geometrical 
Progression. 


No.           Given. 

Requi- 
red. 

Formulas. 

Cor. 

1 

2 
3 
4 

a,     r,    n, 
a,     r,     s, 
a,     n,    s, 
r,     7i,    s, 

/ 

^_a-\-{r-l)s 

r 
l{s  —  ly-'— a{s  —  a)  "-'=  0 
I       {r-l)sr^'-' 
r"—l 

9 
11 

12 
10 

5 
6 

7 

8 

a,     r,    n, 
a,     r,     I, 

a,    n,     /, 

r,     n,     I, 

S 

^__a{r"-l) 
r—1 
rl  —  a 

8 
5 

r—1 

„_   l{r^-l) 
(r-l)r"-» 

GEOMETRICAL    PROGRESSION. 


231 


Requl. 
rod. 


,     n,     /, 


'■)      ^h      *5 

r,     Ij 


_{r  —  l)s 

r" —  1 
=  rl  — (r  —  1)  5 

a(s  —  ay-'—  I  {s — iy-'=  0 


a,     n,    /, 


,        «,       o, 


■J       ")        "3 


n,     /,     s, 


i^^^. 


a  a 


s  —  l 
s 


,r"-'+ -  =  0 

s  —  /  s  —  / 


16 


14 


a,     r,     /, 

a,     /,     s, 
r,     /,     s, 


log/  — log 


+  1 


logr 
log[a+(r— 1>]  — log  a 


log  r 
log  /  —  log  a 


log(s— a)  — log(5— /) 

log  I —  log  [rl — (r — l)s] 

~~  logr 


+1 


20 


18 


(179.)  All  the  formulas  of  the  above  table  are  easily 
drawn  from  the  conditions  of  (28)  and  (32),  which  conditions 
correspond  with  formulas  (1)  and  (5),  except  the  last  lour 
which  involve  logarithms  ;  we  will  hereafter,  under  Loga- 
rithms, show  how  these  formulas  are  obtained. 


232  GEOMETRICAL    PROGRESSION. 

If  in  a  geometrical  progression  we  change  a  to  /,  Z  to  a, 

and  r  to  r~'=-,  the  progression  will  remain  the  same  as 

before,  taken  in  a  reverse  order.  These  changes  being  made 
in  the  formulas  of  the  preceding  table,  we  shall  discover 
that  some  of  the  formulas,  as  in  arithmetical  progression, 
have  correlative  formulas.  Those  having  correlative  for- 
mulas, have  them  referred  to  in  the  table,  under  column 
headed  Cor. 

EXAMPLES. 

1.  The  first  term  of  a  geometrical  progression  is  5,  the 
ratio  4,  the  number  of  terms  is  9.     What  is  the  last  term  1 

Formula  (1),  which  is  /  =  ar"-^^  gives 
/  =5X4^=  327680. 

2.  The  first  term  of  a  geometrical  progression  is  4,  the 

ratio  is  3,  the  number  of  terms  is  10.     What  is  the  sum  of 

all  the  terms  1 

air""—  1) 
Formula  (5),  which  is  5  =  — ,  gives 

s  =  ^\     ^  =  118096. 

3.  The  last  term  of  a  geometrical  progression  is  106f f| 
the  ratio  is  f,  the  number  of  terms  8.  What  is  the  firs* 
term? 

Formula  (9),  which  is  o  =  -j^^-j,  gives 

106^-^ 


(180.)  When  the  progression  is  descending  the  ratio  is 
less  than  one,  and  if  we  suppose  the  series  extended  to  an 
infinite  number  of  terms,  the  last  term  may  be  taken 
^  =  0,  which  causes  formula  6  to  become 


GEOMETRICAL    PROGRESSION.  233 

Which  shoiDS,  that  the  sum  of  an  infinite  number  of  terms 
of  a  descending  geometrical  progression  is  equal  to  its  first 
term,  divided  by  one  diminished  by  the  ratio. 

EXAMPLES. 

1.  What  is  the  sum  of  the  infinite  progression 

l  +  i+T  +  i  +  rV  +  &c.? 
In  this  example  a=  1,  r=  J ,  and  (33)  becomes 

2.  What  is  the  value  of  0.33333  &c.,  or  which  is  the  same 
thing,  of  the  infinite  series  /o-f-  rf  o  "h  t/o  u  ~f~  ^^-  *? 

yV)  and  (33)  gives 


1-A       "       ' 
3.  What  is  fhe  value  of  0.12121212  &c.,  or  which  is  the 
same,  of  j\%  +  r^\%^  +  to  o H o  o  &c.  ? 

In  this  example  a=  j\%,  r  =  j^-^y  and  (33)  gives 


4.  What  is  the  sura  of  the  infinite  series 

i  +  i  +  i  +  .V  +  8VH-&c.? 

5.  What  is  the  sum  of  the  infinite  series. 


30 


Ans. 


Ans.  j. 


234  HAEMONICAL   PROPORTION. 

6.  What  is  the  sum  of  the  series  1-1 |-~5~l~-^  ~h 

X        XT        t' 

&c.,  to  infinity  1 

Ans. 


x—1 


7.  What  is  the  sum  of  the  series  1  -| j-—  -\ 


-f-  &c.,  to  infinity  1 


x+l    '    (x+1)^ 


Ans.  ^±i. 

X 

8.  Suppose  the  elastic  power  of  a  ball,  which  falls  from 
a  height  of  100  feet,  to  be  such  as  to  cause  it  to  rise  0.9375 
of  the  height  from  which  it  fell  ;  and  to  continue  in  this 
way  diminishing  the  height  to  which  it  will  rise  in  geomet- 
rical progression,  till  it  comes  to  rest.  How  far  will  it 
have  moved  1 

Ans.   3100  feet. 

HARMONICAL    PROPORTION. 

(181.)  Three  quantities  are  in  harmonical  proportion, 
when  the  first  has  the  same  ratio  to  the  third,  as  the  differ- 
ence between  the  first  and  second  has  to  the  difference 
between  the  second  and  third. 

Four  quantities  are  in  harmonical  proportion,  when  the 
first  has  the  same  ratio  to  the  fourth,  as  the  difference 
between  the  first  and  second  has  to  the  difference  between 
the  third  and  fourth. 

Thus,  if 

a  :  c  :   :  a  —  b  :  b  —  c,  (1) 

then  will  the  three  quantities  a,  &,  c,  be  in  harmonical  pro 
portion. 

If  a  :  d  :  :  a—b  :  c  —  dj  (3) 


HARMONICAL    PROPORTION.  235 

then  also  will  the  four  quantities  a,  6,  c,  and  d  be  in  harmo- 
nical  proportion. 

Multiplying  means  and  extremes  of  (1),  we  have 

ab—  ac  =  ac  —  be,  (3) 

which  by  transposition  becomes 

ab  -{-bc  =  2ac.  (4) 

In  a  similar  way  equation  (2)  gives 

ac-\-bd:=  2ad.  (5) 

Suppose  a,  6,  c,  d,  e,  &c.,  to  be  in  harmonical  progression; 
then  from  (4)  we  have 

be  -{-  ab  =  2ac,  ") 
cd  +  be  =  2bd,  i  (6) 

de  -{-  cd=  2ce,  } 
&c.  &c. 

Dividing  the  first  of  (6)  by  abc,  the  second  by  bed,  and  the 
third  by  ede,  &c.,  we  find 


(7) 


11111 


From  which  we  sec  that  -,7,-515-)  &c.,  are  in  arith- 
a   b   c    a  e 

metical  progression.     (Art.   165.) 

Hence,  the  reeiproeals  of  any  number  of  terms  in  harmo- 
nical 2iTOgression  are  in  arithmetical  frogression ;  and  con- 
versely the  reciprocals  of  the  terms  of  any  arithmeticar 
progression  must  be  in  harmonical  progression. 


236 


HARMONICAL    PROPORTION. 


when  reduced  to  a 
common  denominator,  are  60,  30,  20,  15,  12,  10,  which  by 
the  above  property  must  be  in  harmonical  progression. 

If  six  musical  strings  of  equal  tension  and  thickness,  have 
their  lengths  in  proportion  to  the  above  numbers,  they  will, 
when  sounded  together,  produce  more  perfect  harmony  than 
could  be  produced  by  strings  of  different  lengths ;  and  hence 
we  see  the  propriety  of  calling  this  kind  of  relation,  har- 
monical or  musical  proportion. 

(182.)  If  we  take  the  arithmetical  mean,  the  geometrical 
mean,  and  the  harmonical  mean,  of  any  two  numbers,  these 
three  means  will  be  in  geometrical  proportion. 
Let  a  and  6  be  any  two  numbers,  then  will 
\{a  -\-  h)=  their  arithmetical  mean, 
y/ah^     "     geometrical     " 
2a6 


a-\-h 
And  we  evidently  have 

\{a  +  h)  :  ^/a6 


harmonical      " 


v/a6 


_2a&_ 
a-\-h' 
That  is, 

The  geometrical  mean,  between  the  arithmetical  mean 
and  the  harmonical  mean  of  two  quantities,  is  the  same  as 
the  geometrical  mean  of  the  quantities  themselves. 


237 


CHAPTER  Vn. 


SERIES. 


METHOD    OF    INDETERMINATE    COEFFICIENTS, 


(183.)  Suppose  we  have  the  following  conditi 


ion  : 


.%  +  A3^  -\-J,x'-\-JlsX^+   &C.    ^  ^j^ 


=Bo+  Bix  +5ox2-f  B3X3+  &c. 
If  the  above  condition  is  true  for  all  values  of  x,  we  must 
have 


^^  =  B.,^  (2) 


X  =B„.  J 
For,  since  the  condition  (1)  is  true  for  all  values  of  x,  it 
becomes,  when  x  =  0,  ^0  =  J^o. 

Now,  rejecting  A^  from  the  left-hand  member  of  (1)  and 
its  equal  ^o  from  its  right-hand  member,  it  wnll  become 
Axx■^Ji<^'^-J^^x'-\-  &c.=i?i.r+7?.jX^+53a-3-|-  &c.      (3) 
Dividing  through  by  x,  we  find 

^i+j?ax-l-^3a:'-f  &c.=^i-f  B.x-^7?3r-^+&c.       (4) 
When  X  =  Oj  equation  (4)  becomes  A\=^B\. 


238 


SERIES. 


(5) 


By  a  similar  process  we  can  show,  that  ^2=1  B^;  Jia^^B^; 
and,  in  general,  ^n  =  Bn^ 

If  we  transpose  all  the  terms  of  the  right-hand  member 
of  (1),  it  will  become 
^o  —  Bo+{A  —  Bi)x-\-{Jiii~~B2)3r  ) 
+  {Jl3— Ba)3^' +  &c.  =  0.  S 

(184.)  Hencey  when  we  have  an  equation  of  the  form  of 
(5),  true  for  all  values  of  x^  it  follows  ^  that  the  coefficients 
of  the  different  powers  of  x,  are  respectively  equal  to  0. 

We  will  now  apply  the  above  principle  in  the  develop- 
ment of  some  particular 


EXAMPLES. 


1  _L.  2x     . 
1.  Required  to  expand  — ^^ — Ij^^^*^  ^^  infinite  series. 

Assume, , 


1-1- 2a: 


A  +  A^x  4-  A-^  -|-^3x'  4-  &c. 


Clearing  this  of  fractions  and  then  transposing,  it  becomes 

I—aAx  —  JIi  Vx-  — ^2>x'-1-&c.=0. 
—  2)    —A)     —a) 
Now,  since  the  right-hand  member  is  equal  to  0,  it  fol 
lows,  by  the  above  principle  of  indeterminate  coefficients 
that  the  coefficients  of  the  left-hand  member  must  each 
equal  0 ;  hence  we  have  the  following  conditions  : 
^0-1=0,         (1)- 
^1—^0  —  2  =  0,         (2) 
^2  —  •^i — -^0  =  0,        (3)1 
.^3  —  ^2-^1  =  0,         (4)>  (A) 


A 


^„_,— ^„_2=0. 


(71  +  1). 


From 

the  above  we 

readily 

find 

(1) 

•^1  =  3, 

« 

(2) 

^2  =  4, 

(3) 

^3  =  7, 

(4) 

(B) 


^„  =  A_i+^,^o,     (n  +  1)/ 


The  value  of  the  general  coefficient  ./?„,  as  given  in  group 
(B),  shows  that,  any  coefficient  is  equal  to  the  sum  of  the 
two  preceding  ones. 


Substituting  these  values,  as  given  by  (B),in  the  assumed 
1  +  2^: 
1  —  X  —  x-^ 


value  01  ' ■  T  we  iind 


1  4-21: 
^  —1 -[-3a: -[-4a:--(- 7x3+ 11x^+18x^-1-  &c. 


1  —  X  —  X- 


2.  Required  the  development  of ; — -  by  this  method . 

1+x+x^ 

Assume 

.,z=.^o  +  ^lX  +  ^2X^  +  ^3x3+^4X*+  &C., 

1+x+x- 
proceeding  as  in  last  example,  we  find 

^0  +  ^1  )      +^.  )       +^'^3)       +^4) 

—  1  )  +jio)    +a;    +^2) 

Equating  the  coefficients  to  zero,  we  have 


240 


SERIES. 


^0  =  0, 

(1) 

^^^^,J—l  =  0, 

(2) 

^2+A+^0  =  0, 

<3) 

^34-^2  +  ^1  =  0, 

(4) 

^4+^3+^2  =  0, 

(5)| 

(B) 


^n+'^n— l+^n-2  =  0. 


(^  +  1). 


Commencing  with  the  first  condition,  we  find  ^o  =  0, 
which  substituted  in  (2)  gives  ^i=:  1,  these  values  of  Jlo 
and  ^1,  substituted  in  (3),  give  ^2  =  —  1,  now  substituting 
jii  and  Ji-i  in  (4),  we  find  ^3  =0,  continuing  in  this  way, 
we  find  Jii=  1  ;  ^5= —  1,  and  so  on;  from  the  general 
condition  (n  +  1)  we  find  jin  =  —  A-i  —  c/?„_2,  that  is, 
any  coefficient  is  equal  to  the  sum  of  the  two  preceding  co- 
efficients taken  with  a  contrary  sign. 


Hence, 


l-\-x-[-x^ 


r-\-x^  —  x^-{-  x'^  —  &c. 


3.  Required  the  development  of  *^1  — x  by  this  method. 

Assume,      v^l  —  x^=JiQ-{-AiX-\-A-ic--{-A^:j?-\-kc. 
Squaring  both  members,  we  find 
1      .        /?  2  .  o  /7   /7    ^     +2A^o  )       +2^o^3  " 

S        +^-;  S     +2A^2 

Equating  like  coefficients,  we  have 

^0-^  =  1,  (1) 

2AA  =  — 1,  (2) 

2AA+^?  =  0,  (3)' 

2A^3+2A-^2  =  0,  (4)( 

2j?oA+2j3iA+^^  =  0.  (5) 


X3+&C. 


(C) 


SERIES.  241 

The  first  condition  gives 

.^o  =  v/l  =  l. 
This  value  of. '7o  substituted  (2),  \ve  find 

2 
In  this  way  we  find,  in  succession,  the  following  values  : 

These  values  substituted  in  our  assumed  value,  give 

^. X       X-        3x^  3.5y^ 

~''~    ~2~~2l~2X6~2.4.6.8~   '^" 
The  general  term  of  this  series  is 
_3.5...(2n  — 3)x" 
2.4.6.8.... 2n    ' 

4.  Required  the  development  of ■ by  this  me- 

^  l—2x-]-x'-    ^ 

thod. 

Ans.   l+3x+5x-^-f7x^-(-9x^-{-lla;5+   &c. 

l-fx 


5.  Required  the  development  of 


1  — X XT 

Ans.    l+2x-f 3z=+5r'+8ar'-|-13x''+  &c. 


(18.5)  Before  closing  this  subject  we  will  develop — 

which  will  be  of  use  hereafter. 
Assume, 

"Ln^  =  ^o-f-./3iy+.%^+ +-^m3/"+  &c.     ( 1 ) 

X — y 

Multiplying  through  by  x  —  y,  we  obtain 
31 


242 


SERIES. 


(A) 

(n  +  1) 
(5) 


—  A  )        -il  )  -'In-l 

Equating  like  coefficients  of  y,  we  get 

yioa:  =  x'^  (1) 

^V  — A  =  0,  (2)1 

^2X  — .'ii^O,  (3)1 

-     Jj^x  —  Jo  =  0,  (4). 

and  in  general, 

.'i„X  — ./?„_!  =0. 

Equating  the  coefficients  of  y",  we  have 

Jl,„X  —  J,n-i=  —  l. 

From  (1),  we  find 

which  substituted  in  (2),  we  find 

.^1  =  x"'-^.  ■ 
This  in  turn,  substituted  in  (3),  gives 

and  in  general  we  have 

JJ^  =  x'"^-\ 

In  this  general  value  of  Jla  write  m  —  1  for  iij  and  we  get 

This  value  substituted  in  (5) ,  gives 

^,„x  —  1  =  —  1,  or  j3,n  =  0, 
and  consequently  all  the  succeeding  values  of  Jin  will  be 
reduced  to  zero. 

These  values  of  j?o ;  -^i;  ^2;  ^3;  &c.,  substituted  in 
(1),  give 


x  —  y 


=  x"^'  -\-x"'--y-Jrx'^Y .  .  .  +xy'"-'^+y'^\     (B) 


SERIES.  243 

BINOMIAL    THEOREM. 

(186.)  We  have  already  found  by  actual  multiplicatioi; 
(Art.  94),  that 
(a+x)'  =  a-|-x, 

(a+x)-  =  a--f2oa;+a:-^,  .  ,^. 

{a-{-xY  =  a^-\-3a-x-{-3ax'+x\  ^  ^    ' 

{a-\-x) '  =  fi  =  4-4a-'a;+6aV'+4ax^-}-a;' 

Now,  the  Binomial  Theorem  teaches  us  the  law  by  which 
we  may  write  the  development  of  {a  -\-  a:)"'  for  any  values 
of  o,  x,  and  //;. 

To  determine  this  law,  assume 

(a-f  x)«  =  .'io +  .iia:  +  . //ox- +  . ^3X^-^1-  *c.         (1) 

We  have  taken  the  exponent  of  this  binomial  fractional, 
in  order  to  make  the  development  more  gen«»^i. 

The  assumed  form  ibr  the  development  of  (a  -f-  a:)"  be- 
ing general,  must  be  true  for  all  values  of  x.     When  a:  =  0, 

m 

it  becomes  o"  =.■?»,  introducing  this  value  oi .'%  in  (1),  we 
have 

(«  -j-  xf  =  «"  -I-  Ax  +  Ax'  -f  At"  -f-  &c.  (2) 

In  (2),  writing  xj  for  x,  and  it  becomes 

(a  -f-  xi)~'  =  (?  +./?ix,  -f-  ^ox^  -j-  ./?3a: f  -j-  &c.         (3) 
Subtracting  (3)  from  (2),  we  find 

{a-]-xY—{a-^XiY=  (    (4) 

If  we  suppose 

H  =  {a+xY;  ui={a-\-XiY,  (5) 


244  SERIES, 

we  readily  find 

m  m 

u'"  —  u{"^  =  (a  4-  x)n  —  {a  +  x.y 


and 


7/1"  =  X JCi. 


(6) 


(7) 


Dividing  the  left-hand  member  of  (4)  by  w" — Wi",  and 
the  right-hand  member  by  its  equal  a:  —  X],  observing  to 

tn  tn 

substitute  u"^  —  Ui"  for  (a-j-a:)«  — (a-j-xi)«"j  as  given  by 
(6),  and  it  will  become 


Wl' 


Ui' 


(8) 


\X  —  Xil  \x X  J  \x  —  xj 

Dividing  both  numerator  and  denominator  of  the  left- 
hand  member  of  (8)  Ity  w  —  i^i,  and  performing  the  divisions 
indicated  in  the  right-hand  member,  and  we  obtain  by  the 
aid  of  equation  (B),  Art.  185,  the  following: 

M'"~'-f-  Miw'"~--|- Wi'"~*  i 

t/,n-'-[_  UyU"-'  -|- «i"-'    ~  >  (9) 

^i-h-^2(x-f-a:i)+^3(x^-fxx,+  x^)-f  &c.  > 

Now,  in  (9),  suppose  a;  =  a;i,  and  consequently  w  =  «i, 
and  it  becomes 


^,-f  2A.iX  -f-  3^3r^-f-  4.i,.r3  -f-  &c.  (10) 


Re-substituting  {a-\-x)n  for  u  in  (10),  and  it  will  become 


(a  -}-  x)n 


^,-f-  2A'a-  +  3^3x'-f  4^4x'-f  &c. 


n   a-f-x 

Multiplying  through  by  a  -|-  x,  and  we  obtain 


(11) 


245 


m 


(a  +  x)"  = 

>        (12) 


+'^i     )    +2^2     )     +3^3 


m 


Multiplying  both  members  of  (2)  by—  and  it  becomes 


n 

m.  m 


(a4-x)"=-a-4--.9ix+-Ji.2xH-8cc.  (13) 
n  n  n  n 

Equating  the  right-hand  members  of  (13),  and  (12),  we 
have 


—  a  «  -| ^\x  -\ A'lX^  -| ^3X-*-f-  ^-c. 

4-  -^1    )  -\-2Ao    )    +3^3    ) 

Now,  by  the  principle  of  (Art.  182),  we  must  equate  the 
coefficients  of  like  powers  of  x,  by  which  means  we  have 

'in 

Ji\a  =  —a" , 
n 

n     ' 

3j?3a  +  2^o  =  -^2, 
11 


pA,a^{p-i)A^,  =  -Aj^u 
n 

or,    a,=.^Jl=:I±Aa^,. 

pa 
From  this  general  value,  we  readily  deduce  the  following 


246 


/»       fn      — I 

71 


^..= 


mim 
n  \n 


1 


A, 


tl-'][l-') 


2.3 


The  general  value  being 


A 


(?-!).;>■ 


2.3.4 


(15) 


These  values  of  Ji\',  Ji-i\  A;  &.C.,  substituted  in  (2), 
we  have 


r .an     -x--|-  &C. 


(a-f-x)  n=a»i-f--.0"      x-f 


If  71  =1,  this  value  of  (A)   becomes 
(a  +  x)-  =  n-  +  7na— 'x  _|_^(^  — ^).  a— ^x^+  &c.     (B) 


If  771=  1,  then  (A)  becomes 


I         i        1     i._, 
(a+ar)n  =  a''H «"      x  + 


n\n 


Gn~a:''  +  &c.  (C) 


The  coefficient  of  the  (j7  +  l)th  term  as  given  by  (15), 
becomes  when  w  =  1 , 

m(;7t— l)(7?i— 2)(w— 3) . . .  .(771— p+2)(77t— p+1) 

2.3.4 (;>  — 1)  -V 


!    (16) 


SERIES.  247 

(187.)  The  numerator  of  this  coefficient  being  formed 
of  factors  decreasing  regularly  by  one,  it  follows  that  when 
p  =  771+1  it  will  vanish,  and  then  the  series  must  termi- 
nate ;  so  that  the  number  of  terms  of  the  expansion  (B) 

will  be  771  -|-  1.     But  when  -    is  fractional,  or  a  negative 
n 

integer,  the  number  of  terms  of  the  expansion    must  be 

infinite. 


When  a  or  X  becomes  negative,  then  those  terms  of  the 
expansion  will  change  signs,  which  contain  odd  poioers  of 
this  negative  quantity. 

(188.)  If  in  (B),  we  write  a  for  a;  and  x  for  a,  we  shall  have 
(a._}_a)  m^  a-'"+??ix'"-  'a  -f  !!i(!!^zi)  a;"'-^-^-  &c.        (17) 


Now,  since  the  left-hand  members  of  (B)  and  (17)  are 
evidently  equal,  their  right-hand  members  must  be  ;  and 
since,  when  m  is  a  positive  integer,  the  number  of  terms 
of  (B)  as  well  as  (17)  is  equal  to  m  -{-  1,  it  follows  that  the 
terms  of  the  expansion  (B)  must  be  homogeneous  and  sym- 


(a  +  a;)"'= 
a'"-f  7na'"-  x^^^^"'~~'^\"'-H''->r. .  .  .  +  max"'-'  +  x^-     ^^^ 

If  in  (A)  we  suppose  a^x=  1,  we  shall  find 


m 


™(^')  ■t-)(^^) 


(E) 
(l  +  l)"=2"=l-f-  4—^-^-1--^ ^-^ +&C 

Tlierefore,  in  any  expansion  of  a  binomial,  whose  terms 
are  both  positive,  the  sum  of  the  coefficients  is  equal  to  the 
same  power,  or  root  of  2. 


248  SERIES. 

(189.)  If  in  (A),  we  suppose  a  =  1  ;  x=  —  1,  we  shall 
have 

^         ^  n  2  2.3  ^ 

That  isy  in  any  expansion  of  a  binomial ^  one  of  whose 
terms  is  negative,,  the  sum  of  the  coefficients  w  =  0 ;  and 
therefore  the  swn  of  the  positive  coefficients  must  he  equal 
to  the  sum  of  the  negative  ones. 

(190.)  By  inspecting  formula  (A),  we  discover  that 
the  coefficients  may  be  found  in  succession  by  the  follow- 
ing 

RULE. 

Multiply  any  coefficient  of  any  term  hy  the  exponent  of 
the  hading  quantity  in  that  ternij  and  divide  the  product  by 
the  exponent  of  the  foil  owing  quantity  diminished  hy  one^  and 
the  result  will  he  the  coefficient  of  the  succeeding  term. 

APPLICATION  OF  THE  BINOMIAL   THEOREM. 

(191.)  We  will  now  make  an  application  of  this  theo- 
rem ;  and,  first,  suppose  in  the  expression  (B),  of  page  246, 
we  make  successively  m=  1,  2,  3,  and  4,  and  the  results 
will  be  precisely  the  same  as  those  first  given  on  page  243. 
If  we  make  in  succession  m^  5,  6,  and  7,  we  shall  obtain 
the  following  results  : 

(a-fx)^=a^-|-5a4x-f-l0aV-j-10aV-|-5ax*+a:«, 
(a+x)«  = 

a'''+6a'*x+15a^xa+20a3a^-fl5aV-f-6ax'^4-x«, 
(a-fx)^  = 

J  a'4-7a«x-|-21aV-|-35a^x8+35aV  1 

I  -f21a=^x«-f-7ax«-i-x^.  S 


SERIES.  249 


EXAMPLES. 


1.  Required  the  expansion  of  (a  -j- rr)  "* . 

In  formula  (C),  make  n  =  2,  and  it  becomes 

{a4-x)  =a  -4- -a     x .a    'x'^-] -.a    'x^ — &c. 

^  ~  '  ~2  2.4  ~2.4.6 

=  a    )  1-1 — a—'^x .  a~-a:--i .a—^x^ — &c.  > 

\    ^2  2.4  ^2.4.6  5 

Writing  the  different  powers  of  a,  which  have  negative 
exponents,  in  the  denominator,  by  which  means  their  expo- 
nents change  signs  and  become  positive  (Art.  49),  and  we 
find 

i       h  (      .    X         x-     .      3x'  3.5x'  ) 

2.  Required  to  expand  (a-f-a;)   . 
Changing  ?i  into  3,  in  (C) ,  and  we  have 

Removing  the  factor  a  ,  and  causing  the  different  powers 
of  a  to  pass  into  the  denominators,  as  in  the  last  example, 
we  obtain 


.A       M.  ,   a:        2a:^    ,     2.5x'  2.5.8x'     ,  ,      ) 

(        3a      3.6a-      3.6.9o^      3.6,9. 12a^  ) 


3.  Expand  (a-(-x)^. 

Making  n  =  4  in  (C),  and  it  becomes 

^  ^  ^  4  4.8  '^4.8.12 

32 


250 
Or, 


SERIES. 


■    4.  Required  the  expansion  of  ( 1 4- a:)  . 

This  example  will  agree  with  example  1,  if  we  write  1 
for  a.     Making  this  change  in  example  1,  we  get 


5.  Required  the  expansion  of  (1  -f-  1)"  or  y/2. 
In  the  last  example  make  a:  =  1,  and  it  becomes 


(1  +  1) 


3.5 


11  3 

"^    ~     "^2       2.4  "^2^4.6       2.4.6.8 


&c. 


6.  Required  the  expansion  of  "^1  —  x  or  (1  — xY • 
In  example  4,  change  x  into  — x^  and  we  get 


(l-^f=l- 


3a:^ 


3.5a;'' 


2.4        2.4.6       2.4.6. 


&c. 


This  expansion  agrees  with  the  one  found  by  indetermi 
nate  coefficients.     (See  Ex.  3,  Page  240.) 

7.  Expand  (a  -\-  x)-^. 

In  (B),  make  m  =  —  4,  and  it  becomes 
(a+x)-' 
=  0-'*— 4a-'*a:  +  lOa-V— 20a-V+  35a-V—  &c. 


_Ml  — —       ]^_^       35x* 


&c.| 


8.  Required  to  expand 


a  -f-2; 


or  (a-f-x)~'. 


SERIES.  251 

Making  m  =  —  1  in  (B) ,  we  find 
—  a-'—  a-'x  -f  a-'->x-—  a-*x'^-{-  a"  V—  a-^x^-\-  &c. 


9,  Required  the  expansion  of  — 


X 

In  the  last  example,  write  1  for  a,  and  — x  for  x,  and  it 
becomes 

1 


1  — a: 


1  _|_  a;  4-  x2+  x^+  x^-{-  x^-\-  x^-{-  &c. 


H,       6  36^  3.763  3.7.116^        .      > 


10.  What  is  the  expa:!sion  of  (a  —  hyi 

b 36-  3.76^ 

"4a       4.8a2      ^.8.12d'       4. 

11.  What  is  the  expansion  of  (a-j-z)    *? 

aH        5^^5.10a^     5.l0.15a3"^5.l0.15.20a^  ^' \ 

12.  W^hat  is  the  expansion  of  (a^ — x)^! 

A         1^,         a:  x^  3x3  3.5x>  ) 

'^  r      2a3       2.4a«       2.4.6a^       2.4.6.8a»^  '  S 

13.  What  is  the  expansion  of  (^  +  7^ — 1)^  1 

If  in  example  2,  we  change  a  into  p,  and  x  into  gv/ZZj 
we  shall  find,  by  reccollecting  that  by  Art.  126,  we  have 


252 


1 

3 


i'^+^A^^-i+lep-Y    3.e.9 


p~^q^\/ — 1 — &C 


14.  What  is  the  expansion  of  {p  —  q^ —  1)^  1 

Changing,  in  example  2,  a  into  p^  and  x  into  —  q  "^  —  1, 
we  easily  find 

J 

{p  —  qV  —  \) 

(192. )  This  theorem  may  be  applied  to  quantities  of  more 
than  two  terms. 

Suppose  we  wish  the  expansion  of  (a-j-6-|-c)'. 
Assume 

and 

(a-f-6-|-cy  =  (a4-d)3. 

Now,  in  (B),  make  x  =  dj  and  m=3,  and  it  will  be- 
come 

{a-\-dy=^-a^-\-3a^d-\-3ad'-{-d\  (1) 

Now,  by  assumption  d  =  h-{-c;  therefore  we  have 

and 

d'  =  b'-\-Zb'c-\-3bc'-^c\ 

These  values  of  d,  d^  and  d^,  being  substituted  in  (1) ,  we 

get 

{a-\-h-\-cy  = 

a3+3a^(6+c)+3a(6^+26c+c^)+&^+36'c+36c2+c3 
_  {  a^4-3a=6-f3a^c+3a6'2-|-Ga6c4-3ac2  ) 
~  I  +63+362c+36c2+c^  S 


SERIES,  253 

We  might  proceed  in  this  May  to  obtain  the  expansion  of 
algebraic  polynomials  of  any  number  of  terms,  but  abetter 
method  will  be  to  deduce  a  multinomial  theorem,  which  may 
be  done  as  follows  : 


MULTINOMIAL    THEOREM. 

(193.)  This  theorem,  as  we  have  just  hinted,  gives  the 
law  of  the  expansion  of 

(aoH-aiX-f-cr22C*-}-a3:c3-{-&c,) " , 

or  of  any  other  polynomial,  having  for  an  exponent  any 
value  whatever.     To  determine  this  law,  assume 

(ao-}-aiX-fa2X--j-&c.)'^'=A+^iX-|-^52X-+&c.  (1) 

When  a:  =  0,  we  have  ao'*  =  -'^o ;  therefore  we  have 

(ao+aix-(-a2a;-H-&c.)^=ao"'+./?ix-f-^o.r^-|-&c.  (2) 

Writing  x^  for  x,  we  have 

(au+aia:i+a2X=4-&c,)^=flo"+^iXi-f-^2X^  +  &c.  (3) 

Subtracting  (3)  from  (2),  we  find 

m 

{ao-\-a^x-\-a2X'-\-kc.)" — (cfo+ai2"i+«2a-^ -f  &c.)"  ^  (4) 

=^i(x  — a:,)4-^2(x3-^x^)4-&c. 

If  we  suppose 

1 
U=z  (flo-f  cix-f  aox--}-&c.)% 

t^i  =  (ao-f-aiXi-j-aaX= -I-&C.)", 
we  readily  find 


254  SERIES. 

f/m_  JJm 

m,  n 

z={ao'^aiX-\~a2X--\-&,c.Y — {ao-\-aiXi-^aiX]-\-  Scc.y 
U»—  C/i^z^rciCx  — a;i)4-a2(x2  — x2)-f-&c. 

Hence  (4)  becomes 

IJm  __XJ^m--  ) 

Jii{x  —  x,)-{-Az{x^  —  x\)-^M^{x^  —  x])+8LC.  \ 


(5) 


Dividing  the  left-hand  member  of  (5)  by  17" —  t/^,  and 
its  right-hand  member  by  its  equal 

ai(x — a;i)-|-aa(x^ — ^\)-\-  &c., 
we  get 

^,{x  —  x^)-+-ji,{^''~-^')-\-M^'—x'i)-^  &c. 

ai(a:  —  xi)-|-aa(x'—x2)-|- 03(3:^—0:?)+  &«• 


(6) 


If  we  divide  both  numerator  and  denominator  of  the 
left-hand  member  of  (6)  by  U — Ut,  it  will  become  [see 
formula  (B),  Art.  185J, 


(7) 


If  we  divide  both    numerator  and    denominator  of  the 
right-hand  member  of  (6)  by  x  —  x  ,  it  will  become 

^i-f  A>(a:-ha:i)H--^3(a^+xx,-|-xi')+  &c.  .g. 

ai-|-a.(x-|-xi)4-a3(a:--ha:xi-f-x^)-|-  &c. 

The   expressions  (7)  and  (8)  are   equal.     Now,  when 
x  =  xi,  the  expression  (7)  becomes 

ml/"'-' 


nU" 


n'  I/"  ' 


which,  by  re-substituting  the  value  of  [7,  becomes 


SERIES. 

m 

m  {ao-j-aix-{-a'23p'j-  &c.)'' 
n  '    ao-\-aix-^aQpc--\-  &c. 

When  x  =  Xij  the  expression  (8)  becomes 
Jli-\-2M2X-\-3Ji3X--{-  &c. 
ai-|-2aaj: -|-3a3X*-f-  ^^c.' 


266 

(9) 

(10) 


Equating  the  expressions  (9)  and  (10),  and  clearing  of 
fractions,  we  have 


-  {ao-{-aiX-i-aox-+kcY  •  {ai-\-2aiX-{-3a:iX''-\-kc)=^  ( 

n  >  (11) 

(oo+aiZ+caxH&c)  .   (^i-)-2.^2^+3.^,x3-}-&c.)  ) 


Multiplying  (1)  by—,  we  have 

-(ao+a,a:+ayX--f&c.)'^=-(.^o+-'?!J^+^-2a;''H-&c.)   (12) 
n  n 

Hence  (11)  becomes 
-(^oH-^ix+.7.^^+&c).(a,-h2a,a-+3a3.T2  +  &c)=  ) 
(«o+a,x+aox-4-&c).(^,-f-2..i.a-+3.^,a:-+&c.)    ) 
By  actual  multiplication  (13)  becomes 


n  hi 


2^oao\ 


2Anu2 

3^00.3 


-X- 4-^-/3^1 
n 


\ni  .,   ,     „ 
— .T^-j-   &c. 


2.^202 
3^lf/3 

=Aia(i-{-2AMf}x-\-^JlzOQX'-\-i:A.\a{^x'^-Y-  &c. 


(1^) 


-'^iffa 


256 


Equating  coefficients  of  like  powers  of  x  in  (14),  we 
have 

n 

2^200-4-^101  =—Jiai4-2 — iou^. 
n  n 

3«43ao-T-2./520i-j-^ia2=  ^ 


4^400+3^301+2^202+^103  = 


-^30i +  2-^200  +  3 ^103  +  4-^004. 

n  '      n  n  n 


If  for  Moj  we  use  its  equal  ao'*?  we  shall  find  from  the  above 
system  of  equations 

m 
>Aq  =  Co"  } 

Ji\  =  — Oo"     Olj 

wi/m        \ 

j3o  =— ^— oo"     o  =  +  -  Co"     Co, 

2  n 

^,J— ^^^ ""    "■' 

f    +- 1    Qo"      aiao  +  -Go"      03. 

V      w  \n         /  w 


257 


These  values  tf  ^o,  .'?,,  ^2,  ^^3,  &c.,  substituted  in  (1), 
we  have 

»n  m  m 

S,7(,7-') 


ffo"      « 1  +  -  Oo"     a.' 


QMM  - 


2.3 


(+"(?-) 


cro"      a? 


a  771    --1 

n 


x^-\- 


'X^  4-  &c. 


(A) 


EXAMPLES. 

1.  What  is  the  cube  of  \-\-x-\-x'^--\-x^-\-x^-\-Si.cA 

If,  in  our  general  expression  (A)  of  the  multinomial  theo- 
rem, we  make  00=^1  =«j=  03=  &c.=  1;  andm  =  3,n=l, 
we  shall  have 

2.  What  is  the  square  root  of  14-a:+a:^+x'4-&c.? 

In  our  general  expression  (A),  we  must  have7n=l,n  =2, 
and  1  =  flo  =  fli  =  «-'  =  f  3  =  &-C. 

3.  What  is  the  cube  root  of  l-f-x-|-a:--|-a:'-f  &c.? 

In  (A),  make  m  =  1,  7i  =  3,  and  1  =  oo  =  ai  =  aj:=:cr5 
.=:&c.,  and  we  get 

(1+x+xH  r'-h&c.)*-l  +  3^+|x»+l^x3  +  &c. 

33 


258 


4.  What  is  the  cube  root  of  1  -\-  -x -}--x'^-{—x^-\-&iC. 


REVERSION  OF  SERIES. 

(194.)  Suppose  we  have 

aiX-\-aox"-\~a2X^-\-aiX^-\-&ic.^y.  (1) 

The  process  by  which  we  find  x  in  terms  of  y,  is  called 
reverting  the  series  (1),  which  may  be  effected  by  the  fol- 
lowing method  : 

Assume 

x=:^iy-}-J].2y--{~Ji3y^-{-^iy'-^Sic.  (2) 

Now,  we  find  by  actual  multiplication,  or  by  means  of 
the  multinomial  theorem, 

x-^  =  ^-;r  +  2^,^2y'  +  2A^3  }  y4_^&e. 

x^  =  ^'y-}-3^]Jl-2y*+  &c. 
x*=^\y'+SLc. 

These  values  of  x,  x-,  a:%  a:"*,  &c.,  substituted  in  (1),  we 
have 

y*+&c. 


^-;a,|+2^,^2aa 


1/3+      ^4ai 

+  2^1^302 
+  3^=1^003 


=  3/-         (3) 


Hence,  we  have  by  the  method  of  indeterminate  coeffi- 
cients, (Art.  182.) 


SERIES.  259 

From  the  above  conditions  we  deduce 

^,— -, 

2a=  —  a,«3 

Jl^  = ; . 


These  values  of./?i,  v^a,  A^^  ^4,  &c.,  substituted  in  (2), 
\ve  have 

1        a„    ,  ,  2a2 — aiQs  , 

-y—  ^!/--l — =— i — y 

l«i       «?  «?  ^  (A) 

5c,^  —  5010003+0704    ,  ,    c. 
—- : y  -\-  «^c. 


So^hat  if  (1)  is  true  for  all  values  of  x  and  y,  then  also 
will  (A)  be  true  for  all  values  of  x  and  y;  and  such  is  the 
general  relation  betwpen  two  series  when  one  is  the  rever- 
sion of  the  other. 

EXAMPLES. 

1.  Given  the  series  a:-J-x'^-}-x'-|-x^-f-&c.=  y,  to  find 
its  reversion,  that  is,  to  find  the  value  of  x  in  terras  of  y. 

Comparing  this  series  vdth  the  series  (1)  of  this  article, 
we  see  that 


260  SERIES. 

1  =  fli=  a_i=  aj^=^ai=  &c., 
these  values  substituted  in  (A),  give 

a;  =  y  — 2/2-}-i/3— y-f  &c. 

2.  Given  x  —  ^x'-{- {x^ '•i^+  &c.=  y  to  find  x. 

In  this  example  Ave  have 

fl  =  1  ;  a.>=  —  i  ;  03— i  ;  04=  —  }  ;  &c., 
which  values  substituted  in  (A),  give 

3.  Given  l-\-x4-~4--—-4-—- 1-  &c.=Vi  to  find  x 

'       '2    '2.3  '^2.3.4^  ^' 

in  terms  of  y. 

In  this  example  we  first  transpose  the  1,  by  which  means 


hav( 


2    '  2.3   '  2.3.4 
This,  compared  with  (1),  we  find 

0^  =  1;   0,  =  ^;   «.=  !  ;   ^^^  =  2^1'   ^'' 
These  values  cause  (A)  to  become 

"       T^   3         4  ' 

or  restoring  the  value  of  y', 

(y-iy^   ,   (y-l)3      (y-ir 
2       "^         3  4 


x  =  {y-l)- 


+  &c. 


4.  Given  X +— -f-  —  -f-'T~^~  "J"^^'^^^' *°  ^"^  ^' 

-CO  4         0 

Ans.  x  =  y—'^-{-^^  —  J~-{-SLC. 
^      2    '^2.3      2.3.4    ' 


261 


DIFFERENTIAL    METHOD. 


(195.)  This  method  shows  how  to  find  any  particular 
term  of  a  regular  increasing  series,  or  the  sum  of  a  certain 
number  of  terms. 

If  we  take  the  regulaV  increasing  series 

oi;  oj;  fla;  «4;  05;  &c.,  (1) 

and  subtract  each  term  from  the  next  succeeding  one,  we 
shall  obtain  the  following  series,  which  we  shall  call  the^r^^ 
order  of  differences  : 

Co  —  d  ;  C3  —  Co  ;  04  —  as;  as  —  04  ;  &c.  (2) 

Again,  subtracting  each  term  of  this  series  from  the  next 
succeeding  term,  and  we  find  for  the  second  order  of  differ- 
ences 
Gs  —  2a-2-\-ai;  04 — 2as-j-a-2;  05  —  2cf4-j-f/3;  &c.         (3) 

Subtracting  again  each  term  of  series  (3),  of  the  second 
order  of  differences,  from  its  next  succeeding  term,  and  we 
get  a  series  of  third  order  of  differences,  as  follows  : 

04  —  3a3-|-3a.>  —  ai  ;  05  —  3fl4-|-3cr3  —  a->;  &c.     (4) 

Subtracting  once  more  we  find,,  for  the  fourth  order  of 
difiFerenccs, 

05  —  4(/4-|-6rt3 — 4ao-f-ai ;  &c.  (5) 

If  we  take  only  the  first  terms  of  the  scries  (2),  (3),  (4), 
(5),  and  represent  them  respectively  by  D,  ;  Do;  D3 ;  D\; 
&C.J  we  shall  have 

Di  =a-2  —  flj. 
Do  =  03  —  2a.. -f-oi, 

D3  =  a4  — 3a3  +  3a2  — fli,  ^  (6) 

D4  =  a3  —  4fl4  -\-6as  —  4a.i+ai,' 
&c 


•262 


The  coefficients  of  the  difTerent  terms  which  constitute 
the  right-hand  members  of  equations  (6)  are  the  same  as  the 
coefficients  of  the  different  terms  of  the  expansion  of  the  bi- 
nomial (1  —  1)'',  whose  expanded  form  is 

n(n— 1)     7i{n—l){n—2)     n{n—l){n—2){n—3) 


l-n+ 


D„ 


2  2.3         '      •       2.3.4 

Hence,  the  general  equation  of  (6)  is 

,  n(7i— 1)  n{n—l){n—2) 

,  fln-f-l nOn-] a?i-l ^-^ On— 2( 

n{n-l){n-2){n-3) 
+ 2-3^4 ««-3  -  &.C. 


•&c. 


(7) 


If  the  terms  of  the  right-hand  members  of  (6)  are  taken 
in  a  reverse  order,  we  shall  have 


Dn= 


When  n  is  an  even  number^ 
„(„_l)(„_2)(,.-3) 

^  2.3.4 


(A) 


When  n  is  an  odd  number.^ 
-a,+na, 5^^a3+-^ :^ -'04/ 


n(n— l)(n— 2)(7?— 3) 
2.3.4 


(B) 


05  +  &C. 


EXAMPLES. 


1.  Required  tlic  first  term  of  the  fourth  order  of  differ 
ences  of  the  series  1,  8,  27,  64,  125,  &c. 

In  this  example  we  have 

ai  =  l-  flo  =  8j  G3  =  27j  04  =  64  j  a5=125and»=4 


SERIES.  263 

These  values  substituted  in  the  formula  (A),  since  n  is  even, 
give 

2.  Required  the  first  term  of  the  third  order  of  differences 
of  the  series  1,  2*,  3^,  4',  &c. 

Ans.  60. 

3.  Required  the  first  term  of  the  fourth  order  of  differ- 
ences of  the  series  1,  6,  20,  50,  105,  &c. 

Ans.  2. 

(196.)   To  find  the  7ith  term  of  the  series 

we  proceed  as  follows  : 

From  the  first  of  the  equations  (6)  of  last  article,  we 
obtain 

flo  =  Gi  +  Di  5 

this  value  of  co  substituted  in  the  second  of  equations  (6), 
gives 

a3=ai  +  2Dx  +  Do; 
proceeding  in  this  way  we  have  the  following  : 

fli  =  flij 

0-2=  fll  -f--Dl5 

fl3=ai  +  2Di  +  D2, 

a4=ai  +  3A4-3Do4-D3,  /"  ^^) 


as  =  Gi  -f  4Di  +  6D,  +  4D3  4-^ 


Where  the  coefficients  of  the  terms  of  the  value  of  On  are 
equal  to  the  coefficients  of  the  terms  of  the  expansion  of  the 
binomial  (1  -j-  I)""',  whose  expanded  form  is 


l  +  (n-l)+^-. 


(„_l)(n-2)    ,   (n-l)(n-2)(n-3) 


2.3 


(n-l)(n^2)(n-3)(n-4)      ^ 
■^  2.3.4  "^ 


(9) 


;  („_i)(,-2)(„-3)    ^^ 

^  2.3 


(C) 


EXAMPLES. 

1.  Required  the  tenth  term  of  the  series 

1,  4,  8,  13,  19,  &c. 

fli  =  l,  4,  8,  13,  19, 

Di  =  3,  4,  5,  6, 

D2=l,  1,  1, 

D3  =  0,  0. 

Hence,  in  this  example, 

ai  =  1 ;  Di  =  3;  Da  =  1  ;  -D3  =  0  ;  and  n  =  10,      " 

which  values  being  substituted  in  (C),  we  find 

9  8 
aio=  1+9.3 H — ~  =  64,  for  the  tenth  term  required. 

2.  Required  the  nth  term  of  the  series  2, 6, 12, 20,  30,  &c. 

ai  =  2,  6,  12,  20, 

i)i  =  4,  6,  8, 

1)2  =  2,  2, 

D3  =  0. 

These  values  substituted  in  (C),  give 

a„  =  2  +  (n-l)  .  4+t:lfc:?l.  2  =  n^+n=n(n+l), 

which  is  the  nth  term  sought. 


SERIES.  266 

3.  What  is  the  7ith  term  of  the  series 

1,  3,  6,  10,  15,  21,  &c.? 

Ans.  !fcti). 
2 

(197.)  To  find  the  sum  of  n  terms  of  the  series 

O]  ;  0-2',  03 ;  04 ;  as;  &c. 

we  operate  as  follows  : 

Take  the  new  series 

0;  fii;  fli+flj;  Oi+cra-l-os;  Ci+ao+a3+a4&c.  (10) 

Subtracting  each  term  from  its  next  succeeding  term,  we 

have 

oi;  0-2 ;  03 ;  04 ;  05 ;  &c. 

which  is  the  same  as  the  original  series  ;  hence,  the  7i-\-l 
difference  of  the  series  (10),  is  the  same  as  the  n  difference 
of  the  proposed  series  ;  therefore,  if  in  the  formula  (C),  we 
change  Oi  into  0,  n  into  n  +  l?  Di  into  Oj,  D)  into  Di,  D3 
into  D2,  &c.,  we  shall  have 

-^       <  .(n-l)(.-2)(n-3)  ^^t 

^  2.3.4.  ^        ) 

which  expresses  the  71  -\-  1th  term  of  the  series  (10),  but  the 

n-\-  1th  term  of  the  series  (10)  is  the  same  as  the  sum  of  n 

terras  of  the  series 

ai  ;  flj ;  03  ;  04 ;  f'a  ;  &c. 

Putting  this  sum  equal  to  »S„,  we  shall  have  for  the  sum  of 

n  terms  of  the  above  scries,  the  following  expression : 

,  n(n-l)  n(n-l)(n-2) 

2  2.6  V  ^D^ 

2T3.4 ^^+^^'- 

34 


^266 


SERIES. 


EXAMPLES. 


1.  Required  the  sum  of  n  terms  of  the  series 
1,  3,  5,  7,  9,  &c. 

ai  =  1,  3,  5j 
i)i  =  2,  2, 

These  values  substituted  in  (D) ,  give 
Sn=^n-{-  n{7i  —  1)  =n  ,  for  the  sum  of  n  terms  sought. 

2.  Required  the  sum  of  7j  terms  of  the  series 
1,  3,  6,  10,  15,  &c. 

ai  =  1,  3,  6,  10, 
A  =  2,  3,  4, 

1)2=1,1, 
i>3=0. 

These  values  substituted  in  (D) ,  give 


Sn  =  n-\-n{7i—  l)-f- 


i{n—l){7i  —  2)_n{7i-\-l){n-j-9. 


2.3  2.3 

3.  What  is  the  sum  of  ji  terms  of  the  series 

1,  2\  3\  4*,  &C.'? 

^"'-  5  +  2  +  3-30- 

4.  What  is  the  sum  of  ti  terms  of  the  series 


Ans.  !ii!L-hi). 
2 


5,  What  is  the  sum  of  n  terms  of  the  series 
1,  2',  3',  4^,  &C.'? 


Ans. 


;n(n-fl))2 


SERIES.  267 

The  answer  of  the  fifth  example,  being  the  square  of  the 
answer  to  the  fourth  example,  it  follows  that 
{ 1  +2+3+4  4-5 -f .  .  .n }  •'=  P+23+3-'+4^'+53+  . .  .n\ 

SUMMATION  OF  INFINITE  SERIES. 

(198.)  Jin  Infinite  Series  is  a  progression  of  quantities 
continued  to  an  infinite  number  of  terms,  usually  according 
to  some  regular  law% 

If  each  term  of  an  infinite  series  is  greater  than  its  prece- 
ding term,  the  series  is  diverging. 

In  general,  when  each  term  is  less  than  its  preceding, 
the  series  is  converging^  but  this  is  not  always  the  case ; 

for  instance,  the  series  1-| —  ~f"  o  H f"  ^^•■)  which  is  called 

a  harmonic  series,  is  not  a  converging  series,  notwithstand- 
ing each  term  is  less  than  its  preceding  one  ;  its  sum  to  in- 
finity is  itself  infinite. 

Ji  neutral  series  is  one  whose  terms  are  all  equal,  and 
their  signs  alternately  +  and  — ,  thus, 

1=  ^  =  a  — a  +  a  — a  +  a  — a  +  &c. 

Mn  ascending  series  is  one  in  which  the  powers  of  the 
unknown  quantities  ascend,  as 

a-{-bx-\-cx^~\-dx^-\-&.c 

A  descending  series  is  one  in  which  the  powers  of  the 
unknown  quantity  descend,  as 

a -|- 6r- ' +- ex-'- -(- (^ar-3  +  &c. , 
t  ^    ,    c    ,    d    ,    „ 


268 


(199.)  If  we  take  the  difference  between  the  two  fractions 


'       '     we  shall  find  ^■ 


pq 


'r+p 


r{r-\-p)      p 


r       f-\-p       'r{r-\-p) 
\r       r-j-pT 


;  hence 


so  that  any  fraction  of  the  form  — — ^ — -  is  equal  to-th  the 
r{r-^p)  p 

difference  between  the  two  fractions  -  and  — ^  :  hence,  it 

r         r-\-p 

follows  that  if  there  be  any  series  of  fractions  of  the  form 

— - — -,  the  sum  of  the  series  will  equal  -th  the  difference 
r{r-\-p)  p 

of  a  series  of  the  form  -  and  another  of  the  form  — ^  :    so 
r  r+p 

that  whenever  this  difference  can  be  found,  the  sum  of  the 

proposed  series  can  be  obtained. 


EXAMPLES. 

1.  Required  the  siim  of  n  terms  of  the  series 

In  this  example  q=  I ;  p  =  l  ;  and  r  takes  successively 
the  values  1,  2,  3, 4,  &c. 


,1+1+1  +  '+. ...1 

^234  n 


=1-^=.- 


.      \2^3^4^         n^n  +  l/y 


If  71  =  00  ,  then  the  sum 


n-fl 


becomes  =  1. 


SERIES.  269 

2.  Find  the  sum  of  ?i  terms  of  the  series 

0+2-3  +  3^  +  4:7  +  ^" 
In  this  example  (7=  1  ;  /^  =  3  j  and  r=  i,  2,  3^4,  &c. 

,,i+^+,i+i+ ' 

1  A         2      J      4  fi 

(1  +  1+ 1+^  +  ^  +  ^V 

\4^5^  n~?i  +  l~7i-f-2~n+3/ 

^1/11 1 1 1_\ 

3\  6       7i-f-l       w+2      7i  +  3/' 

which  becomes,  when  n  is  infinite  — . 

18 

3.  Find  the  sum  of  n  terms  of  —  -j-  —7  -f- \-  &c. 

I4O        0.0        D./ 

Here  we  find  q=  1 ;  p  =  2. 
1,1.  1 


l\    ^3^5^"*'*2n  — 1  '  =:Ul ^—\ 

2;_/l4_l.  _L_4.      1      \t       2\        2n  +  l/- 

(      \3     o"^"**2;i  — l"^27i  +  l/) 

Therefore, 

'^'"  ~2\^~'  2n4^1 ) '  ^"'^  "^^  ^  2  ^^^^^^  '^''  represents 
the  sum  of  n  terms,  and  S^y^  represents  the  sum  of  an  infi- 
nite number  of  terms. 

4.  Required  the  sum  of  the  series 

14.^-|-l_f_  i-L-&c. 
^3^6^  10~ 

This  series  divided  by  2,  becomes 


270 


and  the  sum  of  this  has  already  been  found.  (See  example 
1,  of  this  Art.)  Therefore,  the  sum  of  the  proposed  se- 
ries is 


«"  =  2(l-„-fl);  S-  =  2- 


5.  Find  the  sum  of  the  series  of 
^3.4  5 

3.5 


4 

5.7"^7:9~9ll"^^^' 


/2 
\3' 

-a^+M^ 

}- 

■8- 

7^9 

this  becomes 

1- 

n+1 

2n  +  l 

.n+1 
'2/1+1 

n+1  n  +  1  \; 

"2/1+1  271+3/'. 


(1-1  +  1 


±1-) 


If  we  use  the  upper  sign,  the  quantity  within  the  paren- 
thesis will  =  1  J  if  we  use  the  lower  sign,  then  this  quantity 
will=0. 

Hence,  the  above  expression  will  become 

1      2       1_/1       n  +  1  \      1_       1 
2 


2  ■    n+1       1 

3  2/1+3     2 


3      2 

and  since  p  =  2,  we  find 

12      4(2/1+3) 


1/1       ^  +  1  \     1 
2      \2      2/1+3/     6 


6^2(2/1+3)' 


Srr.= 


The  upper  sign  has  place  when  n  is  even,  and  the  lower 

sign  when  n  is  odd. 

4         4  4  4 

6.  Required  the  sura  of 1 1 1 |-&c. 

^  1.5~5.9~9.13^13.17^ 

J_ 

4n+l 


Ans.   Sn  =  l  —  '^-^;  S^=^l. 


271 


•      7.  Required  the  sum  of  —--{-  —--\- ;r-z-\-  8i.c. 

l.O  .6.4  O.O 


9  9 2^7 

.,.  — 

it  follows  that 


(200.)   Since  ^^^_^^^      ^^+^)('^_j_2^)     ^(^+^)(^,+2;,)' 


r( 


? =:i/_J ? \  (A) 

r+;,)(r+2p)       2p\r{r-hp)     {r+p){r4-2p  j' 


EXAMPLES. 


8.  Required  the  sum  of  ^  +  _|_  +  _|^  +  &c. 

Comparing  these  terms  with  the  fraction  of  the  left-hand 
member  of  (A),  we  discover  that  p  =  1,  and  g  =  4,  5,  6, 
&c.,  andr=  1,  2,  3,  &c. 

1.2'^2.3"^3.4~^ 71(71  +  1) 

_/4_       5_  n+2  n+3         \ 

i2.3'^3.4"^ ?2(//  +  l)"^(//.  +  l)(n+2)j 

1.2^\2.3^3.4^4.5^       n{n-\-l)l      (?i  +  l)(7i+2) 
Now,  by  example  1,  Art.  199,  we  know  that 

J_  ,  1    ,    1  1  -  =  _!i- 

1.2~^2.3"^3.4 ?j(7i-fl)      7?4-l' 

therefore,  the  series  within  the  parenthesis 
_     n      _   I 
"~  H^l       1.2' 


272 

Therefore,  we  have 
4  1     ,      n 


n-f3         _3  I     n^-fn  — 3 


1.2       1.2  '  n+1       (n4-l)(n-|-2)       2  '  (n4-l)(n+2)' 
vvhich,  diyided  by  2p  =  2,  gives 

c— ^   I       n^-\-n  —  3  o     _5 

'*~4~^2(7i+l)(n-f2)'  '^«'-4' 

9.  rindthesumof^^  +  34^+^-A+.-A_+&c. 


^"^-  '^"-4(2;^l~(2n+lX2;i+3))'  "^^^  "l 


10.  Find  the  sun.  of  ^3  +  _|-^  +  34_+^-l_^+&c. 

(201.)  It  is  obvious  that  this  method  must  be  applicable 
to  series,  the  denominators  of  whose  terms  consist  of  more 
than  three  factors  ;  but  our  limits  will  not  allow  us  to  pursue 
this  subject  any  further. 

RECURRING    SERIES. 

(202.)  ^  recurring  series  is  one,  each  of  whose  terms, 
after  a  certain  number,  bears  a  uniform  relation  to  the 
same  number  of  those  which  immediately  precede  it. 

Thus,  the  series 

l-|_2x+8a;-H-28a3  +  100x'  +  35Cx^-f-  &c. 
is  a  recurring  one,  each  of  whose  terms,  after  the  first  two, 
can  be  found  by  multiplying  the  next  preceding  one  by 
3x,  and  the  second  preceding  one  by  2x'-,  and  taking  the 
sum  of  the  products  ;  thus  : 


273 


8x==:2xX3x4-lx2x-, 
28x3=  8x2  X  3x+2x  X  2.r-, 
100x^1=  28x^X3x4-8x^X2xS 
356x*=  100x^x3x  +  2Sx^x2x% 


(^203.)  The  constant  multipliers  3x,  2x-5  taken  together, 
constitute  the  scale  of  relation. 
Suppose  in  general 

to  be  a  recurring  scries  depending  upon  the  scale  of  rela- 
tion ^,  g,  then  we  shall  have 


(A) 


Ai  =  Aij 

(1) 

A-2  =  Ao, 

(2) 

A3^pA.2-{-qAi, 

(3) 

Ai=pA3-\-qA,, 

(4) 

A5  =  pAi-\-qAs,     ■ 

(5), 

An  =  pA„^,-\-qAn-,. 

{n\ 

If  the  scale  of  relation  consist  of  three  parts,  p,  q,  r,  we 
shall  have 


Ai  =  Aiy 

Az  =  Asj 

Ai  =  pAz-\-qAo-\-rAij 
As  =  pAi  -\-  qAz  -j-  r  j3o, 
A  =  pAs  -\-  qAi  -\-  rAsj 


(1) 

(2) 

(3), 

(4)( 

(5) 

(6)1 


(B) 


An  =pAn-\  +  q-^n-l  +  ^*^n-3.         {n )  / 
35 


274 


SERIES. 


And  in  a  similar  way,  the  successive  terms  of  a  recur- 
ring series,  whose  scale  of  relation  consists  of  more  than 
three  parts,  can  be  found. 

If  we  take  the  sum  of  the  group  of  equations  (A),  putting 
Sn  for  the  sum  of  w  terms  of  the  series,  we  shall  have 
S^^,-{-Ji.:r{-p{S—J,—A.)+q  {S—Jn-^—^n)         (1) 

This  solved  for  Sn,  gives 

By  adding  the  group  (B),  and  reducing  as  above,  we  find 

"       I  +q{Sn—A-^'ln-i-Jin)+r{S—A, 
which  solved  for  <S„,  gives 

(H-  q  —l)Jl:  +  {p  —  l)Mo  —  Jij 
5„=/       .        .^  +  :^"^"-^ \A        (D) 


-Jin-l—^n 


+ 


p-rq-\-r  —  i 


(204.)  Proceeding  in  this  way,  we  might  find  similar 
expressions  for  Sn  when  the  scale  of  relation  consists  of  more 
than  three  parts. 

(205.)  If  the  successive  terms  of  a  recurring  series  are 
decreasing,  and  the  series  is  carried  to  an  infinite  number 
of  terms,  the  last  terms  may  be  neglected  in  formulas  (C) 
and  (D),  as  of  no  appreciable  value,  therefore,  supposing 
n  =  00  in  (C)  and  (D),  they  become 


-      p-fq-1      ' 

^  {pJrq—l)Jii-\-{p  — 1)^-2 


A- 


(E) 


•  (F) 


275 


EXAMPLES. 


1.  What  is  the  sum  of  the  infinite  recurring  series 

1  _|_2a:-f-3x-  +  4x^  +  5x'  +  &c.  : 
\he  scale  of  relation  being 

p  =  2x  ;  q  =  —  x^'l 
In  this  example  »/3i  =  1  ;  ^^0  =  20;.     These  values  sub- 
stituted in  (E),  we  find 

e     _(2x— 1)— 2a:^         1 
^        2x  — x^— 1         {l—xY' 

2.  What  is  the  sura  of  the  infinite  recurring  series 

1  +  2x  -f  Sx'  4-  28x-^  +  lOOx^  +  &c  , 
the  scale  of  relation  being 

])  =  3x  ;  q  =  2x'1 
We  also  have  in  this  example  j?i  =  1  ;  ^2=  2a;,  which 
values  cause  (E)  to  become 

s   =      ^-"^ 

"^       1  —  2a:  —  2x' 

3.  What  is  the  sum  of  the  infinite  recurring  series 

1  +  a;  -h  5x2  -f  13^3  _j_  4ia;i  _|_  Scc  ; 

the  scale  of  relation  being 

p=2x  ;  q  =  3x'1 

1—x 


Ans.   Srr,  = 


CO 


1— 2x  — 3x2 


4.   What  is  the  sum  of  the  infinite  recurring  series 
1  _j_  X  -f  2x-^  H-  2x'  +3x'  +  3x^  +  4x«-|-  4x'^  -f  &c.  ; 
in  which  the  scale  of  relation  is 

pz=x  ;  q  =  3^  ;  r  =  —  x"? 


276 


5.   What  IS  the  sum  of  the  infinite  recurring  series 
l_|_4a:4-Gx-4-l]x^-f  28x-*4-63a:^+  &c.  ; 
in  which  the  scale  of  relation  is 
p  =  2x;  q  =  — 


Ans 


(l+x)»- 


•^^-(l+x)^. 


2x2 


3x3 


From  these  examples  >ve  see  that  the  sum  of  an  infinite 
number  of  terras  of  a  converging  recurring  series,  is  in  the 
form  of  a  rational  fraction.  Conversely,  all  rational  frac- 
tions, when  expanded  by  actual  division,  or  by  the  method 
of  indeterminate  coefficients,  as  accomplished  under  Art. 
184,  will  give  a  recurring  series. 

(206.)  When  the  scale  of  relation  is  not  given,  it  may 
be  found  by  means  of  a  few  of  the  first  terms  of  the  series, 
thus  : 

Resuming  our  general  equation  (n)  of  group  (A)  Art. 
203,  where  the  scale  of  relation  consists  of  two  parts,  p 
and  7,  we  have 

^,.=;.A-i  +  7^'.->  (1) 

Writing  ;i-f-l  for  tz,  it  becomes 

A+.-y^A-f-^'l-i-  (2) 

From  these  two  equations  we  readily  deduce 


A- 


M 


(3) 


(4) 


If  in  (3)  and  (4)  we  put  Ji  =  3,  they  will  become 
AxAi  —  A-.'Ai 


P—  A^A,  —  A\  ' 
AiAi  —  Al 


g=— 


A3A1  —  Al 


(5) 
(6) 


SERIES.  277 

From  (5)  and  (6),  we  shall  be  able  to  find  the  scale  of 
relation,  when  it  consists  of  but  two  parts,  by  the  aid  of 
the  first  four  terms  of  the  series.  Equations  (3)  and  (4) 
show  that  the  scale  may  be  found  by  using  any  four  conse- 
cutive terms. 

By  a  similar  process  we  might,  by  the  aid  of  the  first  six 
terms  of  the  series,  find  the  scale  of  relation  when  it  con- 
sists of  three  parts. 

(207.)  A  geometrical  series  may  he  also  a  recurring 
series. 

To  prove  this,  we  will  take  the  general  form  of  a  geo- 
metrical series  (178). 

a  -f  ar  +  ar-  -f  ar^  -\-  ar^  +  ar'  -\- (1) 

Now,  in  order  that  this  may  be  a  recurring  series,  having 
p  and  q  for  the  scale  of  relation,  we  must  have,  (103), 

ar^  =  par  -j-  qa, 
ar^  =  par^  -j-  qar, 
ar*  =  par^  -\-  qar-^ 
ar^=par*  -j-  qar'^^ 


(2) 


ar"  =par'^~^  -\-  qar"~ 

By  striking  out  the  factors  common  to  these  conditions,  we 
see  that  each  becomes 


which  gives 


r'  =  pri-q,  (3) 


_P 


^±i^p^-}-4q.  (4) 


278 


When  these  two  values  of  r  are  real,  and  not  equal, 
there  will  be  two  geometrical  series  which  will  also  be  re- 
curring, having  p  and  q  for  the  scale  of  relation. 

We  will  denote  these  two  values  of  r,  by  r'  and  r" . 
And  since  it  is  immaterial  what  value  we  take  for  a,  the 
first  term,  we  will  take  a'  for  the  first  term  when  we  use  r', 
and  a"  when  we  use  r" .     The  two  series  will  then  become 


a'  _f_  a  V'  +  a  'r'^'  +  a'  r"  -fa'  r"  -\- a'  r' ' -\- . . 


ip) 


(6) 


a"-\-  a"r"-\r  a"r"^-[-  a"r"^+a"r"*-\-  a' V'^+  . . . . ,  ^ 
each  of  which  is  a  recurring  series  having  p  and  q  for  the 
scale  of  relation.  If  we  take  the  sum  of  the  correspond- 
ing terms  of  the  two  series  (5),  we  shall  find 

(u'+a')  +  (aV+a"r")-f(aV^ -■-«"/•"'-) ; 

-\-  i^.'/  '>  a"r"-')  -|-(aV4-f-a"r"')-j-&c., ' 

which  is  not  a  geometrical  series,  but  is,  nevertheless,  a 
recurring  series  having  p  and  q  fpr  the  scale  of  relation. 

In  all  recurring  series  whose  scale  of  relation  consists  of 
two  parts,  we  must,  in  order  to  be  able  to  compute  the 
successive  terms,  know  the  values  of  the  first  two  terms, 
which  we  have  represented  by  Ai  and  Jl-y. 

Since  the  values  of  a'  and  a",  in  (6),  have  not  yet  been 
fixed,  it  is  evident  they  may  be  so  taken  as  to  make  the 
first  two  terms  of  (6)  agree  with  Ai  and  A^.  This  is  effect- 
ed by  making 


a'-fa"=.3.. 

(7) 

aV-|-a'V'=^2. 

(8) 

These  equations  immediately  give 

r'—r"    ' 

(9) 

(10) 

SERIES.  279 

The  nth  term,  Jin-,  of  the  recurring  series  (6)  is 

o'r'''-'-|-o"r"''-^ 

Hence,  if  we  substitute  the  values  of  a'  and  «",  as  given 
by  (9)  and  (10),  we  shall  have 

r'  —  ;•"  r  — r 

(208.)  By  a  similar  train  of  reasoning,  we  might  show 
that  a  recurring  series  whose  scale  of  relation  consists  of 
three  parts,  is  the  sum  of  i\\xee  geometrical  recurring  series, 
having  the  same  scale  of  relation. 


1.  The  recurring  series  l-\-x-\-3x'-\-5x-'-\-  &c.,  whose 
scale  of  relation  is  x  and  2x-,  is  the  sum  of  the  two  follow- 
ing g-eo7?ie^7?ca/  recurring  series,  each  having  the  same  scale 
of  relation: 

3  '3    ^3^3     ~  3        ' 

1       1.1.,      1    ,  ,    1     4      s 

3      3^3         3     ~  3 

1_|_     a;_^3a;^_|_5x3-f-lla;^  +  &c. 

2.  The  recurring  series,  l-j-.r-j-Sx^-f-l^^^'-f-'^c.,  whose 
scale  of  relation  is  2x  and  3x^,  is  the  sum  of  the  two  follow- 
ing gfCome^Wca/ recwm?2o-  series,  each  having  the  same  scale 
of  relation  : 

1.3     ,  9   .  .  27   3  ,81    ,  ,   , 
2"^2''  +  2     "^2"     "^2"'"  + 
1       1      ,   1    ,      1    .  ,    1      ,       . 
2~2'''^2^'~2      "^2  ^  ^^'•'• 
1+    x-f-5x2+ 13x^-1- 41  x'-f-&c. 


280 


3.  The  recurring  series,  l-{-x-\-2x'--\-3x^-\-6x'^-{-kc., 
whose  scale  of  relation  is  2a:,  a;'-,  and  —  2x^j  is  the  sum  of 
the  three  following  geometrical  recurring  series,  each  hav- 
ing the  same  scale  of  relation  : 


-x4--x-  —  -x'^-\ —  X*  —  &c. 
6    ^6  6      ^6 


1+     rr-f-2x--|-3r^+ 6  x^-f  &c. 

(209.)  From  what  has  been  shown,  it  follows  that  we 
may  regard  all  geometrical  series  as  recurring  series,  but 
all  recurring  series  are  not  geometrical  series.  When 
a  recurring  series  is  not  a  geometrical  series,  it  is  the  sum 
of  two  or  more  geometrical  series.  Hence,  a  geometrical 
series  may  be  regarded  as  a  particular  case  of  a  recurring 
series,  the  recurring  series  being  of  a  more  general  form. 

(210.)  We  will  now  give  some  examples  in  which  the 
general  term,  J?,,,  of  a  recurring  series  is  required. 


EXAMPLES. 

1,  Suppose  the  scale  of  relation  of  the  series 
1  +  x  -f  3a;-  +  5r'+  Ux' -\- 21x^ -}-  &c., 
to  be  ^  =  a; ;  g  =  2x^,  what  will  be  the  7ith  term  ? 

In  formula  (A),  the  values  of  r'  and  r"  are  given  by  (4). 
In  this  example  they  become 

r'=2x;  r"=^  —  x. 

From  the  given  series  we  have 

Ji\  =  1  :  »/?2  ==  X. 


SERIES.  281 

Substituting  these  values  in  formula  (A),  vrc  find  the 
Ans.  ^n=  !(2a:)"-'  +  ^(—  x)'^K 

2.  If  the  scale  of  relation  of  the  series 

1-l-x-f  5a:2-}-13x3  +  41x^+121x«+  &c., 
is  ^  =  2x  ;  q  =  3a;"^,  what  will  be  the  nth  term  ? 

Ans.  ^„=i(3x)"-i  +  i(— x)"-'. 

3.  The  scale  of  relation  of  the  series 

1  +  2x  -f  5x*  +  13x3  _|_  34a;4  ^  89x^  +  &c., 
being  j9  =  3x;  q  =  —  x^,  what  will  be  the  nth  term  ? 

(v/5  +  i)(3+  -y^y-'  ^„_, 

Ans.  ^„  =  <f  ^"^^ 

(^5-l)(3-^/5)"-^  ^„_,^ 

Referring  to  the  (question  of  the  oak  tree,  under  Chap. 
XII,  Higher  Arithmetic,  we  see  that  if  we  call  x  =  1,  the 
above  expression  for  Jl„  will  give  the  number  of  branches 
of  the  tree,  at  the  end  of  n  years,  thus  the  number  of 
branches  at  the  end  of  20  years  is 

(1  +  v/5)(3+  v/5)'^ _  (l-v/5)(3-v/5V^ 
2-V5  2-V5 

(211.)  Having  shown  how  to  find  the  general  term  of 
a  recurring  series,  it  is  easy  to  find  the  sum  of  n  terms  by 
the  aid  of  formulas  (C)  and  (D),  Art.  203. 


1.  Find  the  sum  of  7i  terms  of  the  recurring  series 
1-f-x-f  3x2-(-5x3-|-llx'-|-21xS4-  &c., 
whose  scale  of  relation  is 

j)  =  x;  q=2x^ . 
Under  the  last  Article,  we  have  found  the  nth  term  to  be 


282 


Writing  n  —  1  for  n,  we  find 

A-i  =  |(2x)"-H^(-^)"-'- 

Substituting  these  values  of  A  and  A-i?  together  with 
the  known  values  of  p  and  q,  in  (C),  Art.  203,  we  have 
_2''+'{l+x)x^-{  {l—2x){—xY—3 
'^"  ~  d{2x^-i-x  —  l)  ' 

or,  perhaps  a  simpler  form  is 

(2»+^^2)x"+'-f-(2»+'dbl)x''—  3 
^"~~  6x-^  +  3x  — 3 

In  this  last  expression,  the  upper  sign  is  to  be  used  when 
n  is  even. 

2.  Find  the  su-m  of  7i  terms  of  the  recurring  series 
l-{-x-{- 5x-+  13x^-j-41x^ -f  &c., 

whose  scale  of  relation  is 

p  =2x'^  q  =  3r^. 

In  this  example,  the  nih.  term  is 

A=|(3xy-+l(-x)"-', 

or  which  is  the  same  thing, 

A=l(3'-'=Fl)x«-'. 

The  upper  sign  must  be  used  when  n  is  even.     Writing 
n  —  1  for  71,  we  find 

The  values  of  ./?„,  ^„_i,  /j,  and  7,  being  used,  cause  for- 
mula (C)  to  become 

3(3"-'Tl)a:''+'  +  (3"drl)x''4-2j  — 2 
*"""  6x'^-|-4x  — 2 

Use  the  upper  sign  when  n  is  even. 


CONTINUED    FRACTIONS. 


CHAPTER  VIII. 


CONTINUED  FRACTIONS. 


(212.)  Suppose  we  have  the  following  conditions: 
-^=2/+^,  (1) 

^.=2/.  +  g,,  (2) 

D-^=y+^,  (3) 

■Us 

^3=3/3  +  £,  (4) 

&c.  &c. 

In  (1),  for  Di,  write  its  value  as  given  by  (2),  and  it 


becomes  Jl  =  y- 


X.2 


yi-^~.      In    this   expression,    for    Da, 
write  its  value  given  by  (3),  and  we  have 


..+^-^ 


y^'+D, 


284 


CONTINUED    FRACTIONS. 


Now  substituting  for  D3,  its  value  given  by  (4),  and  we 
obtain  j3=y-| — ! 


3/2- 


X3 


J/3- 


X4 


Di-\-  &c. 


(5) 


(213.)  Such  expressions  as  the  above  value  of  ^,  equa- 
tion (5),  are  called  continued  fractions. 


The  expressions  — ,  — ,  — ,  &c.,  of  which  —  is  the  gene- 

Vn 


expressions  — , 

2/1  y-y  3/3' 


ral  term,  we  shall  call  partial  fractions.     The  numerators 
xi,  x-2  X3, a:,i,  we  shall  call  partial  numerators. 

The  denominators  i/i,  y^,  3/35 Vn-,  in  like  manner, 

we  shall  call  partial  denominators. 

If  we  compute  successively  1,  2,  3,  4,  &c.,  terms  of  the 
continued  fraction  (5) ,  by  reducing  them  to  the  form  of 
common  fractions,  we  shall  find 

91' 


1 
1 

yyi+Ji 
3/1 


P3 
9i' 


H 


y^_^^ ^my2_-h^^yr±y^ ^^3^ 

Xi 


3/i-f- 


I  ^3  _yyiy9y3-\-Xiy2y:i-\-yxc^9-\-yyiX3-\-XiX3^ pi 

^       yz                   ^1^23/3+ 3^22/3 +.yi-^8  94* 

&c &c. 


We  shall    hereafter   represent  these  successive   values, 
which  we  shall  call  approximative  fractions .,  by  the  abridged 

expressions  ^,  ^-,  ^,  &c.,  of  which  the  general  term  is  ^. 
qi  q-i  93  qn 


CONTINUED    FRACTIONS.  286 

By  carefully  examining  the  above  approximative  frac- 
tions, we  discover  the  following  relations  : 
p3=p^y2-\-pix,,       (6)  '73=  922/0 +9ta^3,  (8) 

Pi  =  p:iy3-\-p23:3j        (7)  q\=q3y3-^q-2T3.  (9) 

By  referring  to  our  continued  fraction  (5) ,  we  see  that 

—  will  change  to—,  if  we  substitute  3/3-I —  for  y^.     Mak- 
94  qs  y\ 

ing  this  substitution  in  (7)  and  (9),  we  find 

?3|  y^-\ — - )  +i>2X3   v^{pm-\-pi^^)  -\~P^^* 
^_    V      yii         ^ 

93 1  ya  -f  ^  J  +  9--i^3    yi{.q2y-3 + q-^z)  +  93x4 
.p-iy4-|-?3^4 


94yi+!73J^< 


Hence,  ;j5  =  7>4y4+P3^4,  (10)         90  =  904  +  932^4.  (H) 

And  in  the  same  way  may  pr,  and  9,;  be  drawn  from  the 
next  two  inferior  values.  Therefore  the  law  is  general  and 
may  be  expressed  as  follows  : 

Pn  =  Pn-OJn~\-\-Pn~^Xn~:^  (12) 

•      9„=9„_jy,._,-|-9„-ox„_,.  (13) 

(214.)  If  we  place  our  quantities  in  the  following  order  : 

y        y\       y-2       ya       y\ yn~i       yn 

I    .  y_.  Vi   .         Pj.  PJ..  PjlZl   .  Pjl.S^r 

0'        1'        q.'       93'       7.' 9«-i'       qn' 

X\  X  X3  .T4  15  Xn  X.-i-i 

We  may  find  the  successive  approximative  fractions  by 
tke  following 


286 


CONTINUED    FRACTIONS. 


RULE. 

Multiply  the  numerator  of  the  last  approximative  frac- 
tion by  the  partial  denominator  which  stands  over  it,  and 
the  numerator  of  the  approximative  fraction  which  precedes 
this,  by  the  partial  numerator  which  stands  under  it;  the 
sum  of  these  products,  noticing  the  signs,  is  the  numerator 
of  the  next  approximative  fraction.  In  like  manner  we 
must  multiply  the  denominator  of  the  last  approximative 
fraction  by  the  partial  d.enomAnator  which  is  over  it,  and 
the  denominator  of  the  approximative  fraction  which  pre- 
cedes this  by  the  partial  numerator  which  staiids  under  it; 
the  sum  of  these  products  is  the  denominator  of  the  next 
approximative  fraction. 

EXAMPLES. 

1.  Find,  by  the  above  rule,  some  of  the  approximative 
fractions  of  the  infinite  continued  fraction 


9  — &c. 
Our  work,  when  executed  agreeable  to  the  above  rule,  will 
be  as  follows  : 


a 

3 

5 

7 

9 

1 

a 

3a— a'    15 

a — 5a^ — a^ 

105a— 35a-'— 10a 

+a* 

6' 

i' 

3 

5 

15  —  a^     ' 

105  —  10a2 

—  a2 

—  a' 

—  a 

.' 

—  a' 

—  a' 

[&c 

2. 

Find 

some 

of  the  approximate  fractions  of  the 

con- 

tinued  fraction 

CONTINUED    FRACTIONS.  287 


i+?i 


3a 


4a 


14^" 


I4-&C. 

In  this  example,  y,  the  integral  part,  is  nothing,  and  our 
work  is  as  follows  : 

0    111  1  1 

10a         a        a-f-3a-  G-j-7a^         „ 

0'   T'  r   l+2a'   1+5^'    14-9a-f8a--*'     ^' 
a   2a  3a      4a  5a  la 

(215.)  If,  in  our  general  expression  (5),  we  suppose  all 

the  partial  denominators  j/i,  yo,  1/3, y„  to  be  positive, 

and  also  1  =  xi  =  Xo  ==  x^  = =  a:„,  we  shall  then 

have 


yr 


y+'- 


y^-\ (14) 

1/4-}~&C. 

This  is  the  kind  of  continued  fraction  most  commonly 
employed.  Any  common  vulgar  fraction  can  be  converted 
into  a  continued  fraction  of  the  above  form,  by  the  method 
explained  in  my  Higher  Arithmetic^  which  is  equivalent  to 
the  following 

RULE. 

Divide  the  denominator  by  the  numerator ;  then  divide  this 
divisor  by  the  remainder^  and  thus  continue  to  divide  the 
preceding  divisor  by  the  last  remainder,  until  there  is  no 
remainder^  or  until  ice  have  obtained  as  many  terms  as  rce 


288 


CONTINUED    FRACTIONS. 


wish;  then  will  these  successive  quotients  be  respectively  the 

values  of  3/1,  1/2,1/3, yn- 

Note. — In  this  rule  we  have  supposed  the  vulgar  fraction 
to  be  less  than  a  unit,  and  consequently  the  integral  part 
y  =  0  ;  when  the  fraction  is  not  less  than  1,  we  may  first 
reduce  it  to  a  mixed  number,  and  then  proceed  with  the 
fractional  part  agreeable  to  the  above  rule. 

EXAMPLES. 

1.  Convert  4x411 4  into  a  continued  fraction. 


05 

as  Oi  lo 


OJ 


CO  CO 
CO  t^ 
o<  o 


5^1 

y 

II 

CO  0 

Oi  (N 

kO 

CO  CO 

O' 

5»i 

lO  lO 

0 

il 

00  CD 

(N 

Tt^cr. 

lO 

CD  10 

^ 

co"© 

CO 

CO  Tl< 

03 

r-l  O) 

1— 1 

OJ  10 

CO 

CO^CO 

lO 

00 

"* 

CD 

i-i 

^2 

O  CD 


i-i  00 

CO 

0  CD 

CO 

03  l> 

to  c- 

Ol 

iO   ^ 

CD 

/-^ 

00 

CO 

<N 

0< 

O  .1 

(N  I-I 


CONTINUED    FRACTIONS.  289 

Therefore,  41^7  5  = 

o+   — J 

'+— 1 

8+-. 


2.  The  fraction  IHHMf  IK  expresses  nearly  the  ratio 
of  the  diameter  of  a  circle  to  its  circumference  ;  required  to 
expand  it  into  a. continued  fraction. 

Proceeding  as  above,  we  find 
1 

7  + 


1+- 


292+&C. 
(216.)  Referring  to  (14),  we  see  that  Ayy^  since,  in  or- 
der to  obtain  the  true  value  of  ^,  something  must  be  added 

to  y.     Again,  A  <  y-\ — ,  since,  in  order  to  obtain  the  true 

2/1 
value  of  A^  the  denominator  y\  must  be  increased,  and  con- 
sequently y-\ will  be  diminished.     We  can  show  in  the 

same  way  that 

Ayy^- A<:y^- ^ 


3/1  +  -,  y-\ r- 

2/2  ,   1 

3/3 

37 


290 


CONTINUED    FRACTIONS. 


(217.)  Therefore.,  the  value  of  A  is  always  comprised  be- 
tween  two  consecutive  approximative  fractions. 

(218. )  When  \  ==  xi  =  x-,  =  x-  = ^  a:„,  equations 

(12)  and  (13)  become 

Pn=Vn-Xyn--{-pn~;  (15) 

qn  =  7„_i2/n-l +?„--.  (16) 

Multiplying  (15)  by  </„_i,  and  (16)  by  p„_i,  and  then 
taking  the  difference  of  the  results,  we  find 

p^qn-i.  —p„-iqn  =  —  {pn-iq,^-i—pn-2q>,-0-       (l'^) 

This  shows  that  pnq„-i  — Pv—iQn  andp„_i9„_,.  — ;>„_27,i_i 
are  equal  in  numerical  value,  but  contrary  in  sign.  When 
n  =2,  equation  (17)  gives 

Piq~  —ihq2  =  —  {piqo—pi,qi)- 

We  know  that— =-  and— ==-  ;    consequently,  piqo  — 
<?i        1  go      0 

poqi  =  —  1  ;  therefore,  p^qi — piq^z  ~  1,  pnqo  —  p^qs^ — l? 

p^q.t  —  p394  =  1  ;  and  so  on  for  other  similar  expressions. 

Hence,  we  always  have 

PnQn-l  —Pn-\q„  =  ±1-  (l8) 

The  upper  sign  having  place  when  n  is  even,  and  the  lower 
sign  when  n  is  odd. 

If  we  take  the  difference  of  two  consecutive  approxima- 
tive fractions  we  shall  find 


p«7«- 


Pv-\qn 


q„        qn-^  qnqn-i 

By  (18),  we  know  that  the  numerator  of  the  right-hand 
member  of  this  equation  is  =  i  1 .     Hence, 

Pn         Pn-\_     =bl 

q»     qn-i      qnqn-\ 


(19) 


CONTINUED    FRACTIONS.  291 

This  shoics^  that  the  difference  between  any  two  consecu- 
tive approximative  fractions  is  equal  to  the  reciprocal  of 
the  product  of  their  denominators. 

We  have  already  shown  that  the  true  value  lies  between 
the  values  of  any  two  consecutive  approximative  fractions, 

and  since  7„>  qi^-x-,  we  have  -^  > .      Therefore 

the  difference  between  Lj^tL  and  Ji  is  less  than  .  That 

J5,  the  difercncc  between  the  true  value  and  any  approxima- 
tive fraction.,  is  less  than  the  reciprocal  of  the  square  of  its 
denominator. 

dividing  (15)  by  (IG)  we  find 
Pn^]h:-^y^i±Pj^^ 
7„        q„^iy„-i-\~q„^z' 

If,  in  this  ecjuation,  for  y  _i,  we  substitute  the  complete 
denominator,  which    we  will    represent    by    z,   then   will 

"'— =^?.     From  the  form  of  our  continued  fraction  (14), 

it  is  obvious  that  z  will  also  be  in  the  form  of  a  continued 
fraction. 

Thus,  c  =  ^_,  +  ^ 


y.^'- 


y..+1-f 


This  value  of  z  being  substituted  for  y,--.  in  (20),  we 
have  ^=Plil-J-Jjlz:\  (21) 

qn-\Z-\-qn-'2 

Equation  (21)  immediately  gives 

^-J^=  ^"--'?"-^-P"-^?"-^  (22) 

qn-j  9„_i(9,._iz4-7„-o) 

qn-2  5„_2(9„_l2-j-<7„_2) 


292  CONTINUED    TRACTIONS. 

The  condition  of  (18)  causes  these  to  become 


A 


A— 


</„_;(ry„_iZ-f-(/„_o)' 


Pn-- 


(24) 
(25) 


g»-2      i7„-o(9„-iz+g„_o) 

Now,  by  the  nature  of  continued  fractions,  c>l,  and 
q„-\  >  (7„_i:.  Hence,  the  right-hand  member  of  (25)  is 
greater  than  the  right-hand  member  of  (24).  Considering 
the  numerical  value  without  reference  to  the  signs.  This 
shoios^  that  each  appro.iiinafe  fraction  is  nearer  the  true  value 
than  the  preceding  approximative  fraction. 

p. 


If 


■>  A^  conditions  (24)  and  (25)  will  giv( 


Pn-\ 


qn-i 

_Pn-2 


?«-l(7"-l-+?n-2) 


7„_,         q„^o{qn-lZ-{-qn~2) 

If  ?!!ii-<  A,  conditions  (24)  and  (25)  will  give 

qn-2 

A=P-^' - . 

qn-i       7„-i(7„_ir+g„_o) 

A-P^' 


(26) 
(27) 


)• 


qn-2      qn-^iqn-iZ-^-q 
Equations  (26)  and  (27)  give 

ISP^^PjuillyA. 

nqn-2         qn~x) 

Equations  (28)  and  (29)  give 

Hence.,  the  successive  arithmetical  means  of  two  consecu- 
tive approximative  fractions  are  alternately  greater  and  less 
than  the  true  value. 


(28) 

(29) 

(30) 
(31) 


CONTINUED    FRACTIONS,  293 

If  we  take  the  product  of  (26)  and  (27),  we  find,  after  a 
little  reduction, 

_  (p„_ig„_i-  — p„_2?„-2)+(P»-l?»-2— ;j„-2g;,-l)zH-g 

Now,  since  ^—^yjl,  we  have,  by  (18), 

gK-2 

{p,^iqn-2  —  p>,~^qn-l)z-\-Z  =  0. 

Moreover,  we  have  pn—\qn-\Z  yp^-^q^-z. 
Consequently,  ^""'^"~'  >  S'.  (33) 

qn--2qn-l 

Taking  the  product  of  (28)  and  (29),  we  find 


>(34) 


qn-zqn-i 

{p„^^qn-lZ—pn-^n-z)-{-(pn-iqn-2—p„-iq„-l)z-\-Z^ 
■*"  9„_o9„_i(9„_iC-|-9h-2)- 

Now,  since ^^  <  Ji,  we  have,  by  (IS), 

{pn-iq„-2  —  pn-^n-l)z—  Z  =  0. 

And,  as  before,  pn-\qn-\Z  ypn—^n-z- 

Consequently,  ^""'^"'^  <  A^.  (35) 

qn-iq>i-i 

Equations  (33)  oTjd  (35)  show  that  the  successive  geomet- 
rical means  of  two  consecutive  approximative  fractions  are 
alternately  greater  and  less  than  the  true  value. 

(219.)  All  approximative  fractions  are  always  in  their 
lowest  terms.      For  if  not,  let  the  numerator  ;?„,  and  its 


294  CONTIXUED    FRACTIONS. 

denominator  ^-n,  of  the  approximative  fraction—,  have  a  corn- 
s''* 
mon  divisor  h. 

Condition  (18)  shows,  that  if  p,,  and  q^  are  each  divisible 
by  A,  then  its  left-hand  member  must  be  divisible  by  A,  and 
consequently  its  right-hand  member  is  also  divisible  by  h  ; 
that  is,  ±1  is  divisible  by  A,  which  is  absurd  ;  consequently 
it  is  absurd  to  suppose  pn  and  §«  to  be  divisible  by  h. 

(220.)  In  the  case  of  1  =a:i=:a:2  =0:3  = =  ar„, 

the  rule  under  (Art.  214)  will  require  some  modijfication  in 
order  to  appear  in  its  simplest  form.  Thus,  placing  the 
quantities  as  follows  : 


y   3/1   y-i   y-i   y^ y«-i   yn 


y        pi       Pi       Pi  Pn-l 


,f»,&c. 


0      1       92     q:i      Qi  q^x     q„ 

We  deduce  the  successive  approximate  values  by  this 

RULE. 

Multiply  the  mimerator  of  the  last  approxim,ative  frac- 
tion by  the  jio^rtial  denominator  standing  over  it^  and  to  the 
product  add  the  numerator  of  the  preceding  approximative 
fraction^  and  the  sum  will  he  the  numerator  of  the  next  ap- 
proximative fraction.  In  like  manner  multiply  the  denomi- 
nator of  the  last  approximative  fraction  by  the  partial  de 
nominator  standing  over  itj  and  to  the  product  add  the  de- 
nominator  of  the  preceding  approximative  fraction,  and  it 
will  give  the  denominator  of  the  next  approximative  frac- 
tion. 


1.  What  are  some  of  the  approximative  fractions  of  th«- 
iccfinite  continued  fractioii 


CONTINI'ED    FRACTIOyS.  '  295 


3+— 1 

1     2     7     30     157     972 
1'  3'   10'  43'  225'  1393 

2.  What  are  the  approximative  fractions  of  the  continued 

1 


fraction 


3  +  ^ 


1  +  i 


,      1 

1  7  8  23  100  523  623  1769 
3'  22'  25'  72'  313'  1637'  1950'  5537" 
This  last  value  expresses  accurately  the  true  value  of 
the  above  continued  fraction.  Whenever  the  value  of  a 
continued  fraction  is  capable  of  being  expressed  rationally, 
it  must  consist  of  a  finite  number  of  terms  j  but  when  the 
value  is  irrational,  the  continued  fraction  will  extend  to 
infinity. 

(221.)  Continued  fractions  may  be  employed  for  determi- 
ning approximately  the  values  of  the  square  roots  of  surds. 


296 


CONTINUED    FRACTIONS. 


Operating  upon  -v/19,  we  obtain  the  successive  values  : 
'       v/19— 4  3  ^3  ^'^Da 


m  = 


3  v/19  +  2       ,    ,   v/19  — 3  ,    1 

= ■ — =1H — ^=iJc>-\ . 

v/19— 2  5  ^5  ^~~D3 

„  5  v/19  +  3       _,v/19-3  ,1 

„  2  v/19  +  3  v/19— 2  ,     1 

J.        5  vl9  +  2_         v/19-4  1 

3            v/19+4_^  ,   v/19-4_^,    ,     1 
A-  -^^^^^=  -  1 8  + ^  -3'a+^^- 

'       v/19  — 4            3               ^3             ^'^Ds 
&c. 


&c. 


Collecting  the  results,  we  have 
y  =  4,  2/1  =  2,  y3=:l,  2/3=3,   3/4=  1,   2/5  =  2,   3/9  =  8, 
y7  =  2,  &c. 

v^29  +  4         _v/194.2              v/19  +  3 
i^i— g ,  JJi— ,  U3— ,  V4  — 

v/19  +  3    ^  _  v/19+2    J.  _  v/19+4   ^  _  v/19+4 
5~~'^'""        3""'^"-        T"'^'  3~- 

The  values  of  Dj,  Do,  D3,  &c.,  of  which  the  general 
term  is  D„,  arc  complete  denominators  of  their  correspond- 
ing partial  fractions. 

The  values  of  yi,  2/2?  3/35  &c.,  which  we  have  already  call- 
led  partial  denominators^  are  the  greatest  integral  parts  of 


CONTINUED    FRACTIONS.  297 

their  corresponding  complete  denominators  Dj,  D2,  D^,  &c. 

In  this  example,  we  see  that  Dt=  Di  =  - — -^  ,    and 

y7  =  yi  =  2,  hence  the  operation  must  begin  to  repeat  at 
this  point,  and  the  partial  denominators  as  well  as  the  com- 
plete denominators  will  recur  in  an  infinite  number  of 
periods. 

(222.)  Suppose  we  wish  the  value  of  the  surd  ^/5. 
If  a-  is  the  greatest  square  contained  in  5,  with  the  re- 
mainder 6,  we  shall  obviously  have 

'  (36) 

J.  ^  _  1 ^  vAB-U  ^  y/^-f-g  ■ 

^        VB  —  a        B—a"  h 

The  form  of  the  general  value  of  D,i  will  be 

D.=  ^^±^\  (37) 

If  n-\-\  is  written  for  71,  this  becomes 

D„^a^^^+-^V  (38) 

Now,  by  carefully  inspecting  ^the  operations  just  per- 
formed for  finding  the  value  of  v/19,  (Art.  221),  we  draw 
this  relation  : 

^«  =  2/H-^.  (39) 

Substituting  in  (39)  the  values  of  D„,  A.+i)  given  by 
(37)  and  (38),  we  find 

38 


298 


CONTINUED    FRACTIONS. 


This  cleared  of  fractions,  becomes 

B-\-{Mn-\-Mn+l)  VB-j-MnMn+l  ) 

=  ynJrWB  +  ynJrnMn+l-\-jynJyn+l.  ^ 

Equating  the  irrational  parts,  as  well  as  the  rational  parts 
(Art.   116),  we  find 

Mn-{-Mn+l  =  ynJK,  (41) 

B-\-MnMn+l  =  ynJVnMn+l  +  JynJVn+1.  (42) 

These  equations  readily  give 

M„+i  =  ynMn—M„. 

B  —  J^n+l 


M„+l= 


JVn 


(43) 
(44) 


If  in  (44)  we  substitute  for  MnJf-\  its  value  given  by  (43) , 
we  find 

_B-M£ 

Equation  (44)  gives 

j^,  _B-MUi 


yn'-J^n  +  ^ynMn.         (45) 


JVn 


In  (46)  writing  n —  1  for  tj,  we  get 
This  causes  (45)  to  become 


JV;+i  =  K„^t—yn^jr+2ynMn 


(46) 

(47) 

(48) 
1,  J\r„,  and 


Equations  (43)  and  (48)  show,  that  when  JV, 
Mn  are  whole  numbers,  then  will  J^n+i  and  jlf„+i  be  whole 
numbers.  When  n=:  1 ,  we  have  by  (36)  JV„_i=  1 ,  JV^=6, 
and  Mn  =  a- 

Hence,  JV^  and  Mn  are  whole  numbers  for  all  values  ofn. 


(223.)  If  in  (21)  we  substitute 
change  to  v/J?,  and  we  shall  have 


^;_i 


for  c,  ^  will 


CONTINUED    FRACTIONS.  299 

VB  = .  (49) 

Clearing  this  of  fractions  and  reducing,  we  have 

q„^iB+qn-i^I«-i  VB+q»-^^^„-l  ^/5  =pn-i  VB 
+;j„_,JW„_i+p,_,JV;_i. 

Equating  the  rational  quantities,  as  well  as  the  irrational, 
we  have 

Pr^^Mn-,-\-pn-^'n~l  =  Qn-lB .  (50) 

qn-lMn-l  +  g;,-oJV„_i  =  p,^i.  (5 1 ) 

From  (50)  and  (51)  we  readily  deduce 

p„^iqn-2—q„-^Pn-2 

j^^_^_qn-iqn--.B-pn-.,p^,^  ^^3^ 

p„^iqn-2 qn-lpn-1 

It  is  readily  seen,  by  reference  to  (18),  (33),  and  (35), 
that  the  numerators  and  the  common  denominator  of  (52) 
and  (53)  always  have  like  signs. 

Consequently^  J^a  and  Mn  are  positive  for  all  values  ofn. 

Equation  (47)  shows  that  Mn<iVB;  that  is,  J\U  cannot 
exceed  a.  Equation  (43),  by  transposition,  becomes  ynJ^n 
=  Mn+i  -\-Mn  ;  which  shows  that  J^n  as  well  as  y„  cannot 
exceed  2a. 

JVow,  since  the  continued  fraction,  which  expresses  v/B, 
must  extend  to  infinity,  and  since  Mn  as  well  as  JV„  are  posi- 
tive integers  less  than  2a,  it  follows  that  the  values  of  Mn 

and  JVn  in  Dn  =  — — ^—  rnust  recur  in  periods. 


300 


CONTINUED    FRACTIONS. 


Suppose  the  number  of  terms  in  a  period  to  be  w,  so  that 
y„+i  =  2/1,  Mn+i  =  Ml,  JSTn+i  =  JV\.  (54) 

If  z  denote  the  complete  denominator,  corresponding  to 
the  partial  denominator  2/„,  we  must  have 

z  =  y«  — a-f^/5.  (55) 

Hence, 

^j5  ^  PnZ-^Pn-l_Pn{yn  —  a)  +Pn  VB+Pn-l  ^^g. 

q,,Z  +  q,i-i      qniVn  —  a)  -j-  ?»  VB-\-q„^i ' 

Proceeding  with  (56)  as  was  done  with  (49) ,  and  we 
find 

PniVn  —  a)  +Pn-l  =  Bq„.  (57) 

gn(2/n  —  a)  +  qn-l  =  Pn-  (58) 

Equation  (58)  gives 


Pjl  =  .y^-a-{.t=}. 


(59) 


Now,  since  — —  cannot  be  a  whole  number,  it  follows 
5/1 

that  Vn  —  a  is  the  greatest  integral  part  of  ^ ;  but,  since  the 

process  begins  to  repeat  at  this  point,  the  greatest  integer  of 

^  is  a;  therefore,  y„  —  a^a,  hence,  y,i^2a. 

So  that,  whenever  we  obtain  a  partial  denominator  which 
is  equal  to  twice  the  greatest  square  root  of  B,  the  process 
must  begin  to  repeat. 

Equation  (58)  gives,  when  2a  is  substituted  for  y^,, 
qn-i=^Pn  —  o,<l>*j  which,  divided  by  q„,  becomes 

qn      qn 


=P-^-a. 


(60) 


CONTINUED    FRACTIONS.  301 


Inverting  both  members  of  (16),  it  becomes 
1  1 


(61) 


qn  yn-iqn-l  +  qn-2 

Multiplying  the  numerator  of  the  left-hand  member  of 
(61)  by  5,1—1,  and  dividing  the  denominator  of  the  right- 
hand  member  by  the  same  quantity,  we  obtain 


q.-i 


9'^ 

qn~i 

(62) 

Writing  71  —  1  for 

n,  in  (62),  and  it 

gives 

9„_o  _ 

1 

7/„_o-f-(/,._3 

qn-2, 

which  substituted  in 

(62)  gives 

9n-l  _ 

1 

"■       V. 

,   1 

'  -+::  J 

(63) 

Again 

,  in  (62),  writing  n  — 2  for  n 

we  get 

qj^_ 

qn-2 

1 

y„_3-t-9„_4 

q'n-a,  which  substituted  in  (63), 

we  get 

qn-,  _ 

1 

3/n-l 

,        1 

.        1        ' 

yn-2-t- 

1 

!/n-3  + 

qn-4 

(64) 

302  CONTINUED    FRACTIONS, 

Continuing  this  process  we  discover,  that  the  value  of 

i2zL  is  expressed  by  a  continued  fraction,  less  than  a  unit, 

qn 
of  which  the  partial  denominators  are  the  same  as  those  of 
the  continued  fractions  arising  from  ./Bj  tal^  in  a  reverse 
order. 

From  this,  we  see  that  if  the  partial  denominators  are 
symmetrical,  that  is,  of  the  following  form  : 

yi>3/2,J/3, y3,  3/25  2/15 (65) 

then  will  i^=-,    or   q„^i  =  j^,,^   and     conversely,   if 

9n         qn 
qnr~i=pni  then  will  the  partial  denominators  be  symmet- 
rical as  given  by  (65.) 

Now,  (60)  shows,  that  ^^  is  the  same  as  the  value  of 

^  in  the  expansion  of  v//j,  after  a  is  subtracted. 

Hence  it  follows,  that,  if  we  neglect  the  integral  part 
a  =  yj  the  partia/  fJcnominators  of  the  continued  fraction 
arising  from  t.K  ualue  of  ^/B,  will  recur  in  symmetrical 
periods. 

(224.)  We  will  now  extract  the  square  root  of  some  surds 
by  the  method  of  continued  fractions. 

From  equations  (43)  and  (44),  we  readily  find  Jl/„-t-i, 

JV*„^.i,  when  2/„  is  known.     Then  by  condition ^ — ^ 

=  yn+i  +  &c.,  we  readily  deduce  i/n+i-  Continuing  the 
process  in  this  way  we  find  in  succession  all  the  different 
partial  denominators,  of  which  the  general  term  is  y„. 

Observing  the  above  law  of  derivation,  we  have  in  the 
case  of  v/19,  the  following  successive  operations  : 


CONTINUED    FRACTIONS.  303 

^W:"=,„+&c;  M.y,^M..=M.^,,  5^'=^.+.. 
^E±^=4  +  &c;    IX  4-0  =  4;  li=i^=3. 
^=2  +  &c;3x2-4  =  2;l^^=5. 


^^2=l  +  &c;5xl-2  =  3;i^'  =  2. 

^+l=3  +  &o;   2x3-3  =  3;  ^^^^=5. 

:^3=l+&c;   5  Xl-3=2;  1^^'=3. 
&c.,  &c.,  &c. 

These  operations  are  all  so  simple  that  most  of  the  work 
can  be  performed  mentally.  Consequently,  the  conversion 
of  the  square  root  of  a  surd  into  an  infinite  repeating  con- 
tinued fraction,  is  a  very  simple  matter. 

If  jB=2S,  we  shall  have  by  proceeding  as  we  have 
already  done  for  B  =:  19,  the  following  periods  of  partial 
denominators  : 

!/)     2/15     2/2,     2/3,     2/45 

5;     3,     2,      3,    10;    3,   2,   3,    10;   3,  2,   3,   10;   &c. 

and  our  continued  fraction  becomes 

^28  =  5+-— 

3  +  i-, 
.  '+— 1 

3+'- 
2+-, 

3+ 

10  +  &C. 


304 


CONTINUED    FRACTIONS. 


If  we  compute  some  of  the  approximate  fractions,  by 
Rule,  under  Art.  220,  we  shall  find 


5,  3, 
1  5 
0'    1 


37 

7 


10, 
127 
24 


3, 
1307 

247 


&c. 


v/28>5j  ^/28<|;  ^/28>^;  ^28  <^ ;  v/28>^^; 


and  so  on  for  the  successive  values 

from  the  square  root  of  28  by  a  quantity  less  than 


This  last  value  differs 
1 
(247)^' 


In  the  same  way  we  find,  for  the  square  root  of  31,  the 
following  partial  denominators,  the  first  terra  being  always 
the  integral  part  : 

5  ;  1,  1,  3,  5,  3,  1,  1,  10 ;  &c.,  the  approximate  fractions  are 

1    5    6    11    39    206    657    863    1520    16063     . 
_._._. .  .  . .  . .  .  •  5j,c 

0  '  1 '  1 '  2  '  7  '  37  '  118  '  155  '  273  '  2885   ' 
The  square  root  of  44  gives  the  partial  denominators 

6  ;  1,  1,  1,  2,  1,  1,  1,  12;    &c.,  the  approximate  frac- 
tions are 

1    6    7    13    20    53    73    126    199    2514     , 
0  '  1  '  1 '  2  '  3  '  8  '  11  '   19  '  30  '  379  '       " 
The  square  root  of  45  gives 

6;   1,  2,  2,2,  1,  12;  &c., 
1    6    7    20    47     n_4     161    2046 
0  '  1 '  1 '  "3  '  7"  '  17  '  24  '  305  ' 

For  the  square  root  of  53  we  have 

7;  3,  1,  1,  3,  14;  &c., 
q 

&c. 


1    7    22    29    51     182    2599 


0  '  1  '  3  '   4  '  7  '  25  '  357 

(225.)  If  we  suppose  5  to  equal  the  following  infinite 
continued  fraction,  we  have 


CONTINUED    FRACTIONS.  306 

1 


2a-f-- 


^2a+&c.  (66) 

Transposing  a,  and  then  inverting  both  members  of  (66), 
we  have 

1     =2a.'  ' 


2a-\ 

2a' 


2a  +  &c.  (67) 

Adding  a  to  both  members  of  (66),  it  becomes 

s-^a  =  2a-\-^ 


2aA-- 


2a-^^ 


2a  +  &c.  (68) 

Equating  the  left-hand  members  of  (68)  and  (67),  we 

have  s-\-a  ^= j    clearing  of  fractions,  s-  — a^^l  ; 

5  —  a 


.-  .  s=  v/a=^+l,   and 


2a-|-i- 


2a-'  ^ 


2a  +  &c.  (69) 

If  we  make  a  =^  1  in  (69),  we  find 

2+— 

^■^2-f&c.  no) 

If  we  make  a:=2,  we  have 
39 


306 


CONTINUED   FRACTIONS. 


4+- 


4  + 


4+&C. 


(71) 


Again,  suppose 


=  a+- 


2-f- 


2a- 


2+; 


2a4-&c.  (72) 

Transposing  a  and  inverting  both  members,  we  have 


2a-j — 


2+- 


2a-f-; 


2-f-&c.  (73) 

Transposing  2  and  inverting,  we  have 

l_2.-|-2a  ^2_^1 


2a +- 


2-f-&c.  (74) 

Adding  a  to  both  members  of  (72),  and  it  becomes 


^-|-a=2a-| 

2+- 


2a -h 


2+&C. 


(75) 


Equating  the  left-hand  members  of  (75)  and  (74),  we 
have 


s-{-a  = 


l—2s-\-2a 
Clearing  this  of  fractions,  and  reducing,  we  have 


(76) 


2+1 


CONTINUED    FRACTIONS.  307 

1 


2a+— 
2+; 


2a+&c.       (77) 
If  we  take  a  =  2,  in  (77),  we  have 

^6  =  2+L_ 

'+— 1 

2  + 


4+&C. 
Making  a=13,  we  have 

2+- 


v/l82  =  13+       J 


26+^ 

2+; 


26-J-&C. 

(226.)   A  continued  fraction,  and  consequently  any  com- 
mon fraction,  can  be  converted  into  a  series  as  follows: 
Equation  (18)  gives 

72     qi     qxqi 

P3 ya  _  —  1 

93  92  9293 ' 

Pi_Pji_  J_ 

94  93         9394* 


Pn  P„-l_(—  1)" 


9„       7„_i        9„_,9„ 


308 

Hence,  by  addition, 


CONTINUED    FRACTIONS. 


?2=^  + J-. 
qn       qi       qiqz 


qzqa     qaqi 


qn-iqa 


The  terms  of  this  series  continually  decrease,  and  are  al- 
ternately positive  and  negative ;  consequently  the  error 
committed  by  taking  n  terms  of  the  series  is  less  than  the 
(n  +  l)thterm. 


LOGARITHMS.  309 


CHAPTER  IX. 


LOGARITHMS. 

(227.)  Logarithms   are  numbers,  by  the  aid  o-f  which 
many  arithmetical  operations  are  greatly  simplified. 

In  the  following  relations  : 

(A) 


a— 6, 

(1) 

ay  =  c, 

(2) 

a'  =  d, 

(3) 

&c., 

&c. 

X,  y  and  z  are  respectively  the  logarithms  of  6,  c,  and  d. 

(228.)  The  assumed  root  a  is  called  the  base  of  the  sys- 
tem of  logarithms. 

(229.)  If  in  (1),  of  equations  (A),  we  make  x^  0,  we 
shall  have  a*'=:6=  1,  for  all  values  of  a,  therefore  the 
logarithm  of  1  is  always  0. 

(230.)  If  in  (1),  we  suppose  the  base  to  be  negative,  we 
shall  have  ( — a)*=6.  If  b  is  positive,  then  x  must  be 
even,  if  b  is  negative,  then  x  must  be  odd  ;  hence  we  can 
not  represent  all  values  of  b  by  the  expression  ( — a)'. 
Therefore  the  base  of  every  system  of  logarithms  must 
be  positive. 


310 


LOGARITHMS. 


(231.)  If  in  (1),  wfc  suppose  b  to  be  negative,  we  shall 
have  0^=  —  b.     Now,  since  a  is  always  positive,  the  ex- 
pression a^  is  positive  for  all  values  of  x  either  positive  or 
negative. 
Therefore,  the  logarithm  of  a  negative  quantity  is  impossible. 

(232.)  Each  different  base  must  produce  a  different  sys- 
tem of  logarithms ;  the  logarithms  in  common  use  have  10 
for  their  base. 

So  that  we  have 
10°=  1;  101=10;  102=100;  10'=  1000;  &c. 

Hence,  we  have 


log.  1  =  0, 

log.  10=1, 

log.  100  =  2, 

log.  1000=3, 


log. 


log. 


10000  =  4, 
&c. 


10000 
&c.. 


4, 


(233.)  If  we  take  the  product  of  equations  (1)  and  (2) 
of  group  (A),  we  shall  have 

aT+'j—  5c,  (4) 

from  which  we  discover  that,  the  logarithm  of  the  product 
of  two  quantities  is  equal  to  the  s%tm  of  their  logarithms. 

And  in  general,  the  logarithm  of  a  number  consisting  of 
any  number  of  factors  is  equal  to  the  sum  of  the  logarithms 
of  all  its  factors. 

(234.)  It  also  follows  from  the  above,  that  n  times  the 
logarithm  of  any  number  is  equal  to  the  logarithm  of  its 
nth  power. 


LOGARITHMS.  311 

(235.)  If  we  divide  equation  (1)  by  (2),  of  group  (A), 
we  shall  find 

a-^=-,  (5) 

c 

from  which  we  see  that,  the  difference  of  the  logarithms  of 

any  two  quantities  is  equal  to  the  logarithm  of  their  quotient. 

(236.)  We  have  just  shown  that  the  logarithm  of  a 
number  raised  to  the  wth  power,  is  equal  to  n  times  the 
logarithm  of  the  number.  Conversely,  the  logarithm  of 
the  wth  root  of  a  number,  is  equal  to  the  nth  fart  of  the 
logarithm  of  the  number. 

(237.)  We  will  now  show  how  the  numerical  values  of 
logarithms  may  be  found. 

If  X  is  the  logarithm  of  JV  for  the  base  a  we  shall  have 
this  condition  : 

a'=jY.  (6) 

If  we  assume 

we  shall  have 

(l-|-m)*=l  +  n.  (8) 

Involving  both  members  of  this  to  the  yth  power,  we  shall 
have 

(l+^'0'^=(l+^y-  (9) 

By  the  Binomial  Theorem,  we  find 

,    ,  .  xy(xy — 1)      „  ,  xyixy — l)(xy — 2)  „ 

.  \-[-xym-\--^^-^ -■  m^-\-  ^^  ^    ,  '^;^ .m^-\- e^c, 

^  ^    ~       1.2  '  1.2.3  ' 

Equating  these  expanded  values,  rejecting  the  units  of 
both  expressions,  we  have,  after  dividing  through  by  y. 


312 


LOGAUITHMS. 


(       ,  xy—\       ,  ,   {xy—\)  (j-V  — 2)       ,  ,    „      } 
a:J,n_|_..-^.  m-+\J- ^L '.  ,a^J^  &c.  ^  = 

'       2  "^  2.3  ~ 

This  must  be  true  for  all  values  of  y. 

When  2/=  0,  it  becomes 
x[m — Im'^-^-liiv' — |7n'-f-&c.}  =  n  — \n^-\-\n^ — -]7i^-j-&c. 

Hence, 
a:=z:  W.JV=  log.  (14-w)= n-— !-_±_ — i_-J (10) 

Re-substitutins:  a  —  1  for  m,  and  we  have 


log.  (1  +  n): 


n  — ln--\-\n^  — i^^+  &c. 


(11) 


If  we  assume 

1 


M 


(a-l)-Ka-ir+K«-l)'— K«-l)'  +  &c 
we  shall  have 

loc.  {\-\-n)  =  M  \n—^y-+'^n^—\n'+\n^—kc.].    (B) 
If  the  base  be  so  chosen  as  to  render  M=  1,  then  for- 
mula (B)  will  become 


log.  (1+n) 


Ln-  +  in3 


&c.      (C) 


(238.)  The  logarithms  obtained  by  formula  (C)  are 
called  hyperbolic  or  JVapierean,  whilst  the  common  loga- 
rithms given  by  formula  (B),  are  called  Briggean. 

Lord  Napier,  or  Neper,  is  supposed  to  have  first  con- 
structed logarithms.  The  logarithms  in  common  use,  were 
first  calculated  by  Mr.  Briggs. 

(239.)  We  shall  hereafter  denote  the  Napierean  loga- 
rithms by  the  abbreviation  JV*  log.,  whilst  the  common  or 


LOGARITHMS,  313 

Briggean    logarithms  will  be  represented  simply  by  log. 

Hence  formula  (C)  will  become 

JVlog.  (l-j-n)=w  — i7i--f  i;i3_  ,^^,_^^^6_&P     ^Q,^ 

(240.)  By  comparing  formulas  (B)  and  (C)  we  discover 
this  relation 

./J/XJVlog.  (l-fn)  =  log.  (1+n).  (12) 

Therefore, 

*-^Mog.(l+,0  ^^> 

■^=  („-i)-k«-i)Vk°-i)'-&c.' '^ "^"""^  ""^ 

modulus  of  the  system  of  logarithms  whose  base  is  a. 

From  (12)  we  see  that,  the  logarithms  of  any  j^articular 
system  is  equal  to  the  Kapierean  logarithm  multiplied  by 
the  modulus  of  that  particular  system. 

(241. )We  wall  now  proceed  to  calculate  some  Napierean 
logarithms. 

Resuming  formula  (C),  which  is 
JVlog.  (1+n)  =  71  —  ^?!3_^ -in3— i7i4  +  ]n5  — •&c. 
we  have,  when  n  is  made  negative, 

JVl0g.(l  — 7l)=  — 71  — In2— i7l3—  |7l*  — &C.       (1) 

Subtracting  (1)  from  (C),  we  have 

JVlog.(l-f7,)-JVlog.(l-7.)=JVlog.l±^        ^2) 


^■1 


=  2(7l-|-l7l3+J7lS-fl7l^+i7l»-f-&C.) 

If  we  assume  n  =  - — — ,  we  shall  find  -2!^  =  -^X_  ,an(l 
2/>  +  l'  \—n        p     ' 

(2)  will  become 

JVlog.^^=2M-  + L__^._J -f&c.^3) 

^     P  hp+1^3(2p+l)3^5(27>+l)'^^       r^ 

Or,  which  is  the  same  thing, 

40 


314 


LOGARITHMS. 


+  &C. 


JVlog.O;-fl)  = 

If  we  take  p  =  lj  formula  (E)  becomes 


I  (E) 


3^ 


0.66666666 -^  1: 
0.07407407^  3: 
0.00823045 -r  5: 
0.00091449-r  7: 
0.00010161-i-  9: 
0.00001129^11: 
0.00000125-^13: 


:  0.66666666 
:  0.02469136 
:  0.00164609 
:  0.00013064 
:  0.00001129 
:0.00000103 
:  0.00000010 


0.00000014- 
0.00000001 


15  =  0.00000001 


0.69314718  =  JVlog.  2. 
Take  p  =4,  in  formula  (E),  and  we  get 

11 


JVlog.5  =  JVlog.  4+2^^  +  3^ 


+ 


5.9 


^+7:9^+^" 


But  JVlog.5  =  JVlog.  10  — JVlog.2; 

also,  JVlog.  4=2  JVlog.  2. 

Hence,  substituting  these  values  of  JVlog.  5  and  JVlog. 
4,  in  the  above  expression,  and  we  get,  after  transposing, 

JVlog.  10  =  3^1og.2+2Jl+J<^+±,  +  ^,+  &c.| 

Executing  the  calculation,  for  the  sum  of  the  series,  as  in 
the  above  example,  omitting  the  ciphers  on  the  left,  we  ob- 
tain the  following : 


LOGARITHMS. 


316 


0.22222222 -M  =0.22222222 
2469136 
274348-^3  =  91449 

30483 
3387-^5  =  677 

376 
42-^7  =  6 

5 

0.22314354  =  sum  of  series. 
3  JVlog.2  =  2.07944154 


2.30258508  =  JVlog.  10. 

We  are  now  prepared  to  find  the  modulus  of  the  Briggean 
system.  Since  the  base  of  the  Briggean  system  is  10,  and 
the  logarithm  of  the  base  of  any  system  is  1,  we  have  log. 
10  =  1;  formula  (D)  shows,  that  the  common  logarithm 
of  any  number  divided  by  the  Jfapierean  logarithm  is  equal 
to  the  modulus  of  the  common  system. 


Hence, 


=  0.43429448. 


JVlog.  10      2.30258508 

This  value,  when  carried  to  35  decimal  places,  is 

Jkr=  0.43429448190325182765112891891660508. 

We  will  now  proceed  to  calculate  common  logarithms. 

Since  all  numbers  are  either  primes,  or  composed  of  a 
certain  number  of  prime  factors,  and  since  the  logarithm 
of  any  number  is  equal  to  the  sum  of  the  logarithms  of 
all  its  factors,  it  follows  that  it  will  be  necessary  only  to 
calculate  the  logarithms  of  prime  numbers. 


316 


LOGARITHMS. 


By  equation  (12),  Art.  240,  we  see  that  the  Napierean 
logarithm  multiplied  by  JW,  gives  the  common  logarithm. 

Hence, 

log.  2  =  JVlog.  2XM= 

0.69314718  X  0.43429448  =  0.30103000. 

The  logarithm  of  10  we  know  to  be  1,  therefore  the 


log.  5 


log. —=1  — log.  2: 


:  0.69897000. 


Formula  (E),  when  adapted  to  common  logarithms,  be- 
comes 

log.  (p  +  l)=  -) 


^o.-^+^-^l,-^  +  3l2il)3- 


5(2;,  +  l)^ 


&c. 


(F) 


log-(;'+l)  =  log.p-f 

0.86858896  5 -i—-f-—l-—H ! -f  &c.  I 

Take  p  =  2  in  (F) ,  and  we  get 
log.  3  =  log.  2+0.86858896 1  l+^^+_l- +  A_ +&e.  j 


5 

0.86858896 

25 

0.17371779-Hl  .-= 

0.17371779 

25 

694871-^3  = 

231624 

25 

27795^5  = 

5559 

25 

1112-^7  = 

159 

25 

44^9  = 
2 

5 

0.17609 126  =  sum  of  series 
log.  2  =  0.30103000 


0.47712126=  log.  3. 


LOGARITHMS.  ST 

Ifj  in  (F),  we  make  p  =  49,  we  get 
log.  50  = 

log.  49  +  0.86858896ji+3^3+^Lp+&c.  j 

And  since 

log.  50  =  log.  lO-flog.  5j  and  log.  49  =  2  log.  7, 
we  have  by  substitution  and  transposition, 
2  log.  7  = 
log.  10+log.  5-0.86858896  j  i+3-^3+5-^.+&c 

Calculating  the  series,  we  find 

99  0.86858896 
(99)2  ^  9301  877362  -j- 1  =  0.00877362 

89-^3  =  29 


0.00877391  =  sum     of 


log.  5=0.69897000 
log.  10=1. 

1.69897000 
0.00877391 


1.69019609  =  2  log.  7, 
0.84509804  =  log.  7. 

We  might  have  calculated  the  log.  7  by  substituting  6  for 
p  in  (F),  but  the  operation  would  have  been  more  lengthy 
than  the  above. 

The  next  prime  in  order  is  11  ;  to  find  its  log.  we  make, 
in  equation  (F),  p  =  99,  observing  that  log.  100  =  2,  also, 
that 


318 


LOGARITHMS. 


log.99  =  log.  9+log.  ll=2log.  3+log.  11, 
we  thus  obtain 
2  =2  log.  3+log.  11+0.86858896|  ^+  ^^^^ &c.  | 

Or,  by  transposing,  it  becomes 
log.ll=2-2log.3-0.86858896|i^+^^3  +  &c.| 


199 
39C01 


0.86858896 

436477-1-1  =0.00436477 
11-^3=  4 


0.00436481  =  sum  of  series. 
2  log.  3  =r  0.95424252 


0.95860733 


2.00000000 
0.95860733 


3+  &C. 


1.04139267  =  log.  11. 

To  find  the  log.  ofthe  next  prime  13,  we  assume  _p  =  1000 
in  equation  (F),  and  obtain 

log.  1001  = 

log.  1000+0.86858896! 

Now,  since 
1001  =7X11X13,  log.  1001=log.7-j-log.  11+log.  13 

Hence, 
log.  13  = 
3-log.7-log.  11+0.86858896  jA_  +  3--J^^+&c.  J 


2001  ~  3(2001) 


2001 


LOGARITHMS.  319 

0.8685S896 

43407  =  sum  of  series. 

log.  7  =  0.84509804 
log.  11  =  1.04139267 


3.00043407  1.88649071 

1.88649071 


1.11394336=  log.  13. 

We  might  proceed  in  this  way  until  we  should  have  cal- 
culated the  logarithms  of  all  the  prime  numbers  within  the 
limits  of  the  tables. 

If,  in  formula  (F),  we  substitute  q- — 1  for  p^  it  will  be- 
come 

log.  f  =  log.  (,"-  l)+2-Af l^^  +  3j5^,+  &c.  j 

Now,  since  log.  (f  =  2log.  q, 
and         log.  (5=—  1)  =  log.  (<?+l)+]og.  (g  —  1), 
we  have 
log.  (g+l)^21og.<7 

-log.(,-l)-0.86858896J^-^^+3^3+^-j 

When  ?>13,  we  have  this  very  simple  formula  : 

0.86858896      , 
log.(7+l)=2log.(/-log.(9  — 1) 2^irY--     (^) 

This  formula  will  be  true  to  8  places  of  decimals. 

Having  already  obtained  the  logarithms  of  all  numbers 
as  far  as  13,  we  may  now  make  use  of  formula  (G)  for  all 
numbers  exceeding  13,  and  thus  shorten  the  labor. 

(242.)  We  have  already  (Art.  237)  said,  that  the  base  a 
of  the  Napierean  system  of  logarithms  satisfies  the  following 
equation  : 


320 


LOGARITHMS. 


{a-l)-i{a-lfi-l{a-iy-^{a-iy  +  kc.  =  l.  (1) 
From  example  3,  page  260,  we  see  that  if  we  have 


(y-1) 


(3/-l)^  (2/-1? ,  (3/-iy 


-|-&C.: 


(2) 


+  &C. 


(3) 


then  will 

2    '  2.3  '  2.3^4 
Equation  (2)  will  agree  with  (1)  when  y=a,  and  j;=r  1. 


x^       x^ 
3/  =  l+a;+-  +  — +  ; 


Making  these  changes  in  (3),  we  find 

1.1.1   I      1     ,       1       ,        1 
a=  1  +  1+2  +  2:3  +  2:3:4 +2:3X5 

This  series  may  be  summed  as  follows  : 

1 
1 

0.5 

0.16666666 

4166666 

833333 

138888 

19841 

2480 

276 

28 

2 


(4) 


2 
3 
4 
5 
6 
7 
8 
9 
10 
11 


2.71828180  =  base  of  Napierean  logarithms. 
This  value,  when  extended  to  35  decimals,  is  found  to  be 
e  =  2.71828182845904523536028747135266249. 


EXPONENTIAL    THEOREM. 


(243.)  This  theorem  makes  known  the  law  of  the  develop- 
ment of  a^  according  to  the  ascending  powers  of  x. 


LOGARITHMS.  321 

To  determine  this  law,  we  will  assume 

a-  =  l-f.^x  +  7.V+rx'  +  Dx^  +  &c.,  (1) 

both  members  of  which  become  1,  when  x  =  0. 
Changing  x  into  y,  in  (1),  and  we  have 

a^=l  +  .%  +  7?.y-^+Cr  +  Dy+&c.  (2) 

Subtracting  (2)  from  (1),  and  actually  dividing  the  right- 
hand  member  by  x  —  y,  we  obtain 

^     y         _^D{x^-^3^y-i-xy''-\-y^)-\-kcA     ^^ 
Writing  x  —  y  for  x,  in  (]),  and  it  becomes 
a-'^  =  1  +Jl{x  —  y)+B{x  -  y)  -f  C{x  -  yY+kc.    (4) 
Transposing  the  1,  and  multiplying  by  a",  we  get 
ay(a=^-y-l)=ay[Jl{x-y)i-B{x—yy+C{x-yy-^kc.]  (5) 
Dividing  (5)  by  x  —  y,  after  replacing  its  left-hand  mem- 
ber by  its  equivalent  value  a'  — «'',  we  find 


^—^  =  ay[^+B{x-y)-{.C{x-yy-]-D{x-yy-^kc.]{e) 
X — y 
Equating  the  right-hand  members  of  (3)  and  (6),  we 

have 

r^-^B{x-^y)-^C{x'-\-xy-^f)  )  ) 

)  -^D{x'-\-xhj-j-xf-\-y')-\-kc.S  [    (7) 

C  ==a^[JJ+B{x-y)-\-Cix-yY+D{x-yy-^kc.]  ) 
This  is  true  for  all  values  of  x  and  y. 


^_^2Bx4-3Cx=-f4Dx3+&c.  =  a^.^.  (8) 

For  a',  substituting  its  value,  equation  (1) ,  we  find 

A~\-2Bx-\-3Cx'-\-iDx^-\-kc.  )  /gx 

=  A-^A'x-{-^Bx''-{-^Cx^-\-&.c.  S 

Equating  the  coefficients  of  the  like  powers  of  x,  (Art 
183),  we  find 

41 


322 


LOGARITHMS, 


2B  =  A-, 
3C=^w\  Therefon 


\/l  =  .i, 
^-2:3' 


&c.,  &c. 
Hence,  (1)  becomes 


D  = 


,fi' 


2.3.4' 
&c.  &c. 


A-x~ 


A'x' 


It  now  remains  to  find  the  value  of  Ji. 
For   this   purpose,   put  l-|-6^fl,  and   we   have  a=^=: 
(l_j_6")2:j  Avhich,  by  the  Binomial  Theorem,  becomes 

(1  +  6)-  = 

xb  .  x{x  —  l)b^  .  x{x  —  }){x  —  2)b' 
~^  T"^        1.2       "^  1.2.3 


+&c. 


(11) 


Performing  the  multiplications    indicated,  we   find   the 
coefficient  of  the  first  power  of  x  to  be 

1       2^3        4^        ' 
or,  re-substituting  a  —  1  for  6,  it  becomes 


Therefore, 


^=(a  — 1)  — 


{a-iy    {a-iy  _  (g-iy 

ft         '        1"  O  A 


+  &c.   (12) 


If  in  formula  (C'),  we  put  a  —  1  for  7J,  we  shall  find 

jv,„,.„=(„_j)_(i_-zI)'+(i:=:ll'-(^Vc, 


LOGARITHMS. 

323 

Hence, 

j3=JVlog.  .'/. 
This  value  of  A  substituted  in  (10),  gives 

(13) 

ar^l+jVlog.  a.a:+l ^_^_  +  ^ ^^_ 

-+  (A) 

When 

a=  e  =  2.7182818  &c., 
then 

• 

./Ylog.  a=./Vlog.  e=  1, 
and  (A)  becomes 

e^=l+x+-  +  _+iL.+  &c.         (B) 


APPLICATION    OF    LOGARITHMS. 

(244.)  By  the  aid  of  a  table  of  logarithms,  we  can  easily 
perform  the  following  operations  : 

1 .  Find  the  value  of  ^-^ — —^ —  by  logarithms. 

Recollecting  (Art.  233)  that  the  logarithm  of  the  product 
of  several  factors  is  equal  to  the  sum  of  their  respective 
Jogarithms  ;  and  (Art.  235)  the  logarithm  of  the  quotient 
of  one  quantity  divided  by  another  is  equal  to  the  logarithm 
of  the  dividend  diminished  by  the  logarithm  of  the  divisor, 
we  find  for  the  logarithm  of  our  expression 

log.  — '^l^i^  =log.  3.75+log.  1.06  — log.  365. 

By  the  tables  we  have 

log.  3.75  =  0.5740313 
log.   1.06  =  0.0253059 

0.5993372 
log.  365=2.5622929 


log.  0.01089  =  2.0370443. 


324  LOGARITHMS. 

Therefore,  the  above  expression  is  nearly  equal  to 
0.01089 

2.  Finil  the  11th  root  of  11,  that  is,  the  value  of  the  '^11. 
Taking  the  logarithm  of  this  expression,  we  find 

locr.  iVn  =  TVof  log.  11  = -'-of  1.0413927  =  0.0946721 

=  log.  1.24357  &c. 
Therefore,  ^  11  =  1.24357. 

3.  What  is  the  value  of  ^"  ^  Y  '  '? 

V6 

5  X  log.  8-f-  ?f  log.  7  - 1  log.  G  =  4.51545  4-0.2816993 

—  0.1556302  =  4.6415191  =log.43794.53. 
Therefore,  our  expression  is  equivalent  to  43804.53. 

(245.)  The  above  exaniples  will  show  the  great  advantage 
of  logarithms  in  abridging  arithmetical  labor.  In  the  higher 
parts  of  analysis,  the  use  of  logarithms  is  indispensable.  It 
would  not  be  difficult  to  propose  questions,  which  by  loga- 
rithms might  be  wrought  in  a  few  moments,  but  if  wrought 
by  arithmetical  rules,  would  require  years.  The  following 
example  will  illustrate  the  above  remark. 

How  many  figures  will  be  required  to  express  9^^  1         w 
The  exponent  of  the  above  expression  is 

90  =  387420489         .  • .  999  =  9^ « '  *  2  0  4  s  9 

Putting  it  into  logarithms,  we  have 

log_  9 3 87. 2 04 89^ 387420489 X log.  9  = 

387420489x0.954242509439=  369693099.63  &c. 

Hence,  the  number  answering  to  this  logarithm  must  con- 
sist of  369693100  figures.  This  number,  if  printed,  would 
fill  upwards  of  256  volumes  of  400  pages  each,  allowing  60 
lines  to  a  page,  and  60  figures  to  a  line. 


LOGARITHMS.  325 

EXPONENTIAL    EQUATIONS. 

(246.)  An  exponentiiil  equation  is  one  where  the  unknown 
quantity  enters  as  an  exponent. 

Thus,  a^=h;  x^  =  c;  &c., 

are  exponential  equations. 

(247.)  When  the  equation  is  of  the  form  a'^  =  b,  we  find, 
by  taking  the  logarithm  of  both  members,  xX  log.  a=log.  6. 

Therefore,  a::^:^  j-Sli'. 

log.  a 

(248.)  When  the  exponential  is  of  this  form  a;*=c,  we 
must  find  the  value  of  x  by  the  following  double  position 

RULE. 

Find  by  trial  two  numbers  as  near  the  value  of  x  as  possi- 
blcy  and  substitute  them  successively  for  x;  then,  as  the  dif- 
ference of  the  results  is  to  the  diference  of  the  two  assumed 
numbers,  so  is  the  difference  of  the  true  result,  and  either 
of  the  former,  to  the  diference  of  the  true  7iumber  and  the 
supposed  one  belonging  to  the  result  last  used-,  this  differ- 
"ence,  therefore,  being  added  to  the  supposed  number,  or  sub- 
tracted from  it,  according  as  it  is  too  little  or  too  great, 
will  give  the  true  value  nearly. 

^ind  if  this  near  value  be  substituted  for  x,  as  also  the 
nearest  of  the  first  assumed  numbers,  unless  a  number  still 
nearer  be  found,  and  the  above  operations  be  repeated,  we 
shall  obtain  a  still  nearer  value  of  x;  and  in  this  way  we 
may  continually  approximate  to  the  true  value  of  x. 

EXAMPLES. 

].   Given  x^=^  100,  to  find  an  approximate  value  of  z. 
The  above  equation,  when  put  into  logarithms,  becomes 


326  LOGARITHMS. 

xXlog.  a-  =  log.  100  =  2.  (1) 

By  a  few  trials  we  find  the  value  of  x  to  fall  between  3 
and  4.  If  we  substitute,  in  succession,  3  and  4  in  (1) ,  we 
shall  find 

3  X  log.  3  =  1.4313639 

4  X  log.  4  =  2.4082400 


0.9768761  =  diff.  of  results. 
0.9768761   :  1   :  :  0.4082400  :  0.418. 
Hence,  4— 0.418  =  3.582  =  x  nearly. 

Upon  trial,  this  value  is  found  to  be  rather  too  small,  whilst 
3.6  is  rather  too  great ;  therefore,  substituting  each  of  these 
in  succession  in  (1),  we  find, 

3.582  X  log.  3.582  =  1.9848779 
3.6  X  log.  3.6=2.0026890 


0.0178111  =difF.  of  results. 
0.0178111   :  0.018  :   :  0.0026890  :  0.002717. 
Hence, 

3.6  — 0.002717  =  3.597283=0;  nearly. 

2.  Given  x"  =z  5,  to  find  an  approximate  value  of  x. 

Ans.  x=2.1293. 

3.  Given  a:*  =  2000,  to  find  an  approximate  value  of  a: 

Ans.  x  =  4.8278. 

4.  Given  X^'  =  100,  to  find  an  approximate  value  of  x. 

Ans.  rr  =  2.2127. 

COMPOUND  INTEREST  AND  ANNUITIES  BY  LOGARITHMS. 

(249.)  Interest  is  money  paid  by  the  borrower  for  the  use 
of  the  money  borrowed. 


LOGARITHMS.  327 

It  is  estimated  at  a  certain  rate  per  cent,  per  annum;  that 
is,  a  certain  number  of  dollars  for  the  use  of  $100  for  one 
year. 

The  sum  upon  which  the  interest  is  computed  is  called 
the  principal. 

The  principal,  when  increased  by  the  interest,  is  called  the 
amount. 

When  the  interest  of  a  given  principal  is  paid  at  the  end 
of  each  year,  it  is  called  simple  interest ;  but  when  the  in- 
terest due,  at  the  end  of  each  year,  goes  to  increase  the  prin- 
cipal, it  is  called  compound  interest. 

The  present  worth,  at  compound  interest,  of  a  given  debt, 
due  at  some  future  time,  is  such  a  sura  as,  being  put  out 
at  compound  interest,  will,  in  the  given  time,  amount  to 
the  debt. 

Jin  annuity  is  a  fixed  sum  which  is  paid  periodically,  for 
a  certain  length  of  time. 

(250.)  In  our  calculations  we  shall  use  the  following  no- 
tation : 

^  =  the  principal. 
r  =  the  interest  of  $1  for'one  year. 
ii  =  $l-{-r  =  the  amount  of  $1  for  one  year. 
a  =  the  amount  of  the  given  principal. 
j3  =  an  annuity. 

a'  =the  amount  of  a  given  annuity. 
P  =  the  present  worth  of  a  given  annuity. 
n  =the  time  in  years. 

Since  $l-|-r=  7?  is  the  amount  of  $1  for  one  year,  it 
follows,  that  the  amount  of  a  given  principal,  /),  will  in  the 
same  time  be  pR,  and  this  being  considered  as  a  new  prin- 
cipal, will  in  the  next  year  amount  to  pR  X  R  ^=^pR^:  which. 


328 


LOGARITHMS. 


in  turn,  will  the  next  year  amount  to  pR^xR  =pR^;  and 
so  on. 


Hence, 


pR  =z  amount  for  1  year. 
jj7{*  =  amount  for  2  years. 
pR^  =  amount  for  3  years. 
pjR4  _.  amount  for  4  years. 


pR'^  =  amount  for  n  years. 
Therefore,  we  have  this  relation, 

w^hich,  in  logarithms,  becomes 

log.  a  =  log,  7J-|-7i  log.  7?.  (1) 

(251.)  When  an  annuity  is  left  unpaid  for  n  years,  it  is 
obvious  that  the  annuity  due  at  the  end  of  the  first  year,  must 
be  on  interest  n  —  1  years,  and  must  therefore  amount  to 
AR"^^ ;  the  annuity  due  at  the  end  of  the  second  year  will 
be  on  interest  n  —  2  years  and  will  therefore  amount  to 
JlR'^'^j  and  so  on,  hence,  the  amount  of  the  annuity  at  the 
end  of  n  years  will  be 

J?(/J»-4-il»-2+ -R+1). 

The  geometrical  progression  within  the  parenthesis  being 
summed,  we  have,  after  substituting  r  for  R  —  1, 

.'=^(^).  (.) 

We  have  said  that  the  present  worth  of  a  debt  is  such  a 
sum  as  being  put  out  at  interest,  will,  in  the  given  time, 
amount  to  the  debt,  hence  we  have 


PR^ 


i'^} 


(2') 


LOGARITHMS.  329 

from  which  we  find 

P=tiL^,  (3) 

When  the  annuity  is  continued  forever,  the  value  of  n 
becomes  infinite,  making  this  substitution  in  (3),  we  find 

The  amount  of  $1  at  compound  interest  for  n  years  at  r 
per  cent.  J  is 

(l  +  r)».  (5) 

The  amount  of  $1  at  simple  interest  for  n  years  at  r  per 
cent.,  is 

l-\-nr.  (6) 

Expression  (5),  when  expanded  by  the  binomial  theorem, 
becomes. 

r 

(l+r)"=  l  +  nr+  -^__^-^  +  ^ ^ -V'+&c. 

When  ri=  1,  this  expression  becomes 

l-\-nr. 
When  n>l,  this  expression  is  >1-|- nr. 
When  7i<l,  it  is  <l-}-;2r. 

Hence,  the  compound  interest  computed  by  formula  (1), 
is  equal  to  the  simple  interest  when  the  time  is  one  year  j 
it  is  greater  than  the  simple  interest,  when  the  time  is  greater 
than  one  year,  and  less  than  the  simple  interest,  when  the 
time  is  less  than  one  year. 


42 


330  LOGARITHMS. 


1.  How  much  will  $875  amount  to  in  12  years,  at  6  per 
cent,  compound  interest  ? 

In  this  example,  we  have 

p  =  S15',  n=12;  R=  1.06; 
and  we  are  required  to  find  a. 

Substituting  these  values  in  equation  (1),  we  have 

log.  a=log.  875 -f  12  log.  1.06. 
By  actually  consulting  a  table  of  logarithms,  we  find 

log.  875  =  2.9420081 
12  log.  1.06=0.3036708 

log.  a  =  3.2456789. 
Therefore,  a  =  $1760.672. 

2.  What  principle  will,  in  10  years,  at  5  per  cent.,  amount 
to  $1000? 

By  transposition,  equation  (1)  becomes 

log.  p  =  log.  a  — 71  log.  R. 
Substituting  for  a,  n  and  R  their  given  values,  we  have 
\og.p=z\og.  1000— 10  log.  1.05, 
.-.  ,og.p  =  3  —  0.2118930  =2.7881070. 
And,  2)=  $613,913. 

3.  At  what  rate  per  cent,  will  $100  in  16  years  amount 
to  $160? 

Fquation  (1)  gives 

log./i='.2SiJ?-!£iiP, 

n 
which,  in  this  example,  becomes 


LOGARITHMS  331 

„      :2.2041200  — 2      ^^,„^.^. 
log.  R= ; =0.0127575. 

.-.  il=  1.02981. 
Therefore,  the  per  cent,  is  2.981,  or  nearly  3  per  cent. 

4.  In  how  many  years  will  $460  at  7  per  cent.,  amount 
to  $1000  ? 

Again,  equation  (1)  gives 

log.  a  —  log.  p 

which,  in  this  example,  becomes 
3  —  2.6627578 


?l: 


11.477  years,  nearly. 


0.0293838 

5.  What  is  the  amount  of  an  annuity  of  $200,  which 
has  remained  unpaid  14  years,  at  6  per  cent.,  compound 
interest  ? 

Equation  (2),  when  put  into  logarithms,  becomes 
log.  a'=log.  ^-}-^og-  (^" — 1)  —  log.  r. 
In  the  present  example 

r  =  0.06  ;  i?  =  1.06  ;  .^  =  200  ;  n=  14. 

log.  R"  =  71  log.  R  =  0.3542826. 

.-.  72"  =  2.2609  and  /{"—  1  =  1.2609. 

Hence, 

log.  a'=  2.3010300-1-0.1006806  —2.7781513, 
and 

log.  a'=  3.6235593. 
Therefore, 

a'=$4203  nearly. 

6.  What  is  the  present  worth  of  the  above  annuity  ? 
Equation  (3)  gives 

log.  P=:  log.  a'  — nlog.  R. 


332 


LOGARITHMS. 


In  this  particular  case,  we  have 

log.  P  =  3.6235593  —  0.3542826  =  3.2692767. 
and  P  =$1858.988. 


7.  What  is  the  present  worth  of  an  annuity  of  $100,  to 
continue  forever,  at  7  per  cent  1 

A 
r 


By  equation  (4),  which  is  P  ^  — ,  we  find 
„      $100 


0.07 


=  $1428.571. 


8.  A  debt,  due  at  the  present  time,  amounting  to  $1200, 
is  to  be  discharged  in  seven  yearly  and  equal  payments. 
What  is  the  amount  of  one  of  these  payments,  if  the  inter- 
est is  calculated  at  4  per  cent  1 

In  this  example,  we  have  given  the  present  worth  of  an 
annuity,  the  time  of  its  continuance  and  the  rate  of  interest, 
to  find  the  annuity. 

Equation  (2'),  by  a  slight  reduction,  becomes 

jR"— 1' 
which,  in  logarithms,  is 

log.  ^  =  log.  P-f  log.  r-f-n  log.  i2  — log.  (/?»--l). 
If  we  take 

P  =  $1200;  r  =  0.04;  21=1.04;  7i  =  7, 

we  shall  find 

.^  =  $199,931. 

9.  In  what  time  will  a  given  principal,  at  compound  inte- 
rest, amount  to  m  times  the  principle  1 

Under  example  4,  wc  have  the  formula 

^og-  a  — log.p 

**-        log.  iJ— ' 


LOGARITHMS.  333 

To  make  tins  agree  with  the  present  case,  we  must,  for  a, 
write  mp,  by  which  means  it  becomes 

log.  m 

(252.)  When  the  interest,  instead  of  being  added  to  the 
principal  at  the  end  of  each  year,  is  added  at  any  other 
regular  period,  as  half  yearly,  quarterly,  &c.,  n  must  be 
considered  as  standing  for  the  number  of  those  periods,  and 
r  will  be  the  interest  for  one  of  those  periods. 

10.  What  is  the  amount  of  $100,  for  three  years,  at  6 
per  cent,  per  annum,  when  the  interest  is  added  at  the  end 
of  every  6  months  l 

Equation  (1),  when  adapted  to  the  present  example, 
becomes 

log.  a  =  log.  100  +  6  log.  1.03, 
from  which  we  fmd 

a  =$119,405 

11.  If  the  interest  of  $1,  for  the  crth  part  of  a  year,  is 
-  ,  what  will  be  the  amount  of  $1  for  ?i  years,  when  a*  =  00  1 

X 

The  formula  for  the  amount  will,  in  this  case,  be 


-;)■ 


Expanding  the  right-hand  member  by  the  Binomial  Theo- 
rem, we  find 

•  ,        r  ,  nx{nx—V)    r^  ,  nx(nx—l)(nx—2)  r^  ,  „ 

a=l-t-nx.— J \— — '.    ^-\ 5: il i^    ,-h&c- 

X  1.2         x^  1.2.3  x»^ 

When  X  =  CO,  this  becomes 

^     ^1.2^1.2.3^1.2.3.4^ 


334 


LOGARITHMS. 


Comparing  this  with  formula  (B),  Art.  243,  we  have 

a  =  e"'. 
Using  the  common  logarithms,  we  have 
log.  a  =/)rX 0.4342944819. 

This  formula  gives  the  logarithm  of  the  amount  of  $1  for 
n  years  at  r  per  cent. 

If  we  call  n  =  1 ,  r  =  0.07,  we  shall  find  1 .0725  nearly,  for 
the  amount.  That  is,  the  instantaneous  compound  interest 
of  $1  for  1  year,  at  7  per  cent,  per  annum^  is  7  j  cents  nearly. 
Hence,  however  often  a  person  adds  the  interest  to  the  prin- 
cipal, to  form  a  new  principal,  he  cannot  make  more  than 
1{  per  cent.,  when  the  rate  is  7  per  cent,  per  annum.  If 
we  take  the  rate  per  cent,  per  annum,  at  6|^,  and  compute 
the  instantaneous  compound  interest  on  $1  for  1  year,  it 
will  be  just  the  same  as  the  simple  interest  of  $1  for  the  same 
time  at  7  per  cent. 

(253.)  Before  closing  this  chapter,  we  will  show  how  for- 
mulas 17,  18,  19,  and  20  of  Geometrical  Progression  were 
found. 

By  taking  the  logarithm  of  both  members  of  No.  1,  as 
given  in  the  table  under  Art.  178,  we  have 

log.  I  =log.  a-\-{n  —  l)log.  r. 

This  gives 

n  — i=!£Hiizzl^ii? 

log.  r        ' 

lop;.  / —  log.  a  .   , 

or,  n=-li- ^-+1, 

log.  r 

which  agrees  with  No.  17. 

No.  5  is  readily  put  in  the  following  form  : 

(i-\-{r —  l)s  —  ar^. 


LOGARITHMS.  336 

Taking  the  logarithms,  we  have 

log- [«  +  ('"— 1)5]  =  log.  a-f-nlog.r, 
from  which  we  readily  get 

„^  Iog-[fl  +  (^  — 1)^]— log-" 
log.  r  ' 

which  agrees  with  No.  18. 

No.  12  may  take  the  following  form  : 

a[s  —  aY~'^  =  l{s  —  /)"-', 
which,  in  logarithms,  is 

log.  a+(7i — l)log.  (5— a)  =  log.  /•+■(" — l)log.(5 — O5 
which  gives 

log./  —  log.  a  , 

^-log.(,_a)-log.U-/)"^   ' 
which  agrees  with  No.  19. 

Again,  No.  16  may  be  written  as  follows  : 
,.(.,_/^,— 1  __.„.«-!  _}_/  =  0, 
which  is  readily  reduced  to 

[r/—(r—l  »"-!  =  /. 
Taking  the  logarithms,  we  have 

log.  [rl  —  (r  —  l)5]+(?i — 1)  log.  r^log.  I. 
From  this  we  find 

_  log.  /  —  log,  [rl  —  (r  —  1  )s] 


1  +^' 

log.r 


which  is  the  same  as  No.  20. 


336 


GENEBAL  PROPERTIES  OF  EQUATIONS. 


CHAPTER  I. 


GENERAL  PROPERTIES  OF  EQUA- 
TIONS. 


(254.)  Any  number  or  quantity  which,  when  substituted 
for  the  unknown  quantity  in  an  equation,  satisfies  the  equa- 
tion, is  called  a  root  of  that  equation. 

If  the  general  algebraic  equation, 
a;'»+j3ix'^^+^.>c"-2 ....  +A,,-,x+An  =  0,  (1) 

IS  satisfied  by  making  a:=ai,  then  a^  is  a  root  of  equa- 
tion (1) . 

Substituting  a^  for  a;  in  (1),  we  get 
aV+Aar'^+'^acr'-  •  •  •  -}- A-id -|-^n  =  0.  (2) 

Subtracting  (2)  from  (1),  we  have 

^„.  x(x  — aO=0.       (  ^^ 

We  know  that  each  of  the  expressions 
x»  — ay, 
a:"-*  — ay-', 
x"-=  — ap=, 

x--ai, 


GENERAL  PROPERTIES  OF  EQUATIONS.        337 

is  divisible  by  a:  —  Oi  ;  consequently,  the  left-hand  member 
of  (3)  is  divisible  by  x  —  a,. 

Equation  (3)  does  not  ditfer  from  (1),  since  (3)  was  de- 
rived from  (1)  by  subtracting  from  it  equation  (2),  which 
is  equal  to  0.  Therefore,  equation  (1)  is  also  divisible  by 
X  —  Ci  ;  hence  the  following  property  : 

(255.)  If  ai  is  a  root  of  the  general  algebraic  equation 

z"+J?,x"-'+^2a;»-2 -\-Jn-ix-\-Jln  =  0, 

then  its  left-hand  memher  will  be  divisible  by  x  —  aj. 
As  an  example,  suppose  3  is  a  root  of  the  equation 

a:3  _  7x2  _f  36  =  0. 
Now,  by  the  above  property  this  equation  must  be  divisi- 
ble by  x  — 3. 

Actually  performing  the  division,  we  have 


x3_7x-+36 
x3  — 3x2 


X  —  3  divisor. 

X- — 4.T —  12  quotient. 


—  4x2 

—  4x2-1- 12x 

—  12x+ 36 

—  12x-h36 

0. 
(256.)    If  we  divide  our  general  equation 

x»+^iX"->  +  ^.^'-2....-f-^„_,x+A  =  0,  (1) 
by  X  —  flfi,  we  shall  obtain  for  a  quotient,  a  new  equation 
of  one  degree  less  than  equation  (1),  which  may  be  repre- 
sented as  follows  : 

X»-'+5,X»-2-f-B2X"-3  ....  -|-5„_oX  +  J?„_i  =  0.  (2) 

This  equation  must  also  have  a  root,  which  we  will  rep- 
resent by  cg.     Again,  dividing  (2)  by  x  —  a^,  we  shall  obtain 
43 


338  GENERAL    PROPERTIES    OF    EQUATIONS. 

a  new  equation  one  degree  less  than  (2),  and  consequently 
two  degrees  less  than  (1).     Let  this  new  equation  be  rep- 
resented by 
arn-s-f  Cix"-3  -f  Cox'-' +  C„_sx4-  C„_o  =  0.     (3) 

If  fl;  is  a  root  of  equation  (3),  we  can  divide  it  by  x  —  03, 
we  shall  thus  find  a  new  equation  of  three  degrees  less  than 
equation  (1).  If  we  continue  in  this  way,  we  shall,  after  n 
divisions,  obtain  an  equation  whose  degree  =  0  ;  therefore, 
equation  (1),  is  composed  of  n  factors. 

X  —  flj  ;  X  —  fl.j  ;  X  —  03;  &c. 

Hence,  we  have  the  following  property  : 

(257.)  If  til,  a-2,  Oiiy a„,  denote  the  n  roots  of  our 

general  equation  of  the  nth  degree^  then  this  eqtiation  will 
take  the  following  form  : 

{x  —  ai){x  —  ai){x  —  ai) {x  —  a;,_i)(x  —  c„)  =  0. 

This  equation  is  verified  by  making  either  of  the  n  fac- 
tors =  0  ;  that  is,  by  making  x  =  Ci,  or  x  =  a^,  or  x  =  a^, 
&c.,  from  which  we  infer,  that  every  equation  of  the  nth 
degree,  has  n  roots. 

(258.)  It  does  not  however  follow,  that  all  the  roots  Oj, 
02}  035  "4,  &c.,  are  different,  since  two  or  more  of  them  may 
be  equal,  but  still,  their  number  must  be  7i,  since  there  are 
n  factors. 

(259.)  If  all  the  roots  C],  oo,  03, an  are  negative, 

then  each  factor  of  the  equation 

(x+a,)(x+rto)(x-fa3) (a:+a„-i)(a:-f  fln)  =  0, 

will  be  positive,  consequently  each  term  of  its  equivalent 
value 

x»-|-^ia:'-'  +  ^,x'^= _|_^^_^3,_|_^_  _0 

•will  be  positive. 


GENERAL    PROPERTIES    OF    EQUATIONS.  339 

If  the  roots  are  all  positive,  tlien  will  the  terras  of 

(x  —  ai){x  —  a.2){x  —  03) {x  —  a,.-i)(x  —  a„)  =  0, 

when  expanded,  be  alternately  positive  and  negative. 

(260.)  HmcCy  if  the  terms  of  any  equatioii  are  neither 
all  positive^  nor  alternately  positive  and  negative^  that  equa- 
tion must  contain  both  positive  and  negative  roots. 

(261.)  Reasoning  after  this  manner,    Harriot    has  shown 

That  every  equation  whose  roots  are  possible,  has  as  many 
changes  of  signs  from  -{-to  —  ,  or  from  —  to  -\-,as  there 
are  positive  roots;  and  as  many  continuations  of  the  same 
signs  from  -f-  ^o  -f-j  or  from  —  to  —  ,  as  there  are  nega- 
tive roots. 

(262.)  If,  as  we  have  already  supposed,  the  n  roots  of 
an  e  , nation  of  the  7ith  degree  be  denoted  by  ^i,  a^,  Qa, . . . .  a,,, 
we  can  put  the  equation  under  the  following  form  : 
(x  —  a^){x  —  a.i){x  —  03)(a: — a^) (x  — a„)  =  0.       (!) 

Let  us  suppose  «;>«.. ;  a,>03;  ajya,i;  and  so  of  the  rest. 

If  a  quantity  b  greater  than  «i  be  substituted  for  x  in 
(1),  the  result  will  be  positive,  since  all  the  factors  w-ill 
then  be  positive. 

If  a  quantity  c  less  than  ci,  but  greater  than  Qo,  be  sub- 
stituted for  a-,  the  factors  will  be  all  positive  except  one, 
and  consequently  the  result  will  be  negative. 

If  a  quantity  d  less  than  ag,  but  greater  than  a.),  be  sub- 
stituted for  X,  all  the  factors  except  two  will  be  positive  ; 
and  since  two  negative  factors  produce  a  positive  product, 
the  result  must  be  positive. 

By  following  out  this  plan  of  reasoning,  we  deduce  the 
following  property  : 


340  GENERAL    PKOPCUTIES    OF    EQUATIONS. 

(263.)  if  tu-0  qucnitilics  he  successively  substituted  for  x 
in  any  equation,  and  give  results  ajfected  with  different 
SIGNS,  there  must  be  an  odd  number  if  roots  heticeen  these 
quantities. 

But  if  the  tico  quantities  when  substituted  for  x  give 
results  affected  with  the  same  signs,  there  must  he  either 
no  roof,  or  else  an  even  number  of  roots  between  these  quan- 
tities. 

EXAMPLES. 

1.  Find  the  first  figure  of  one  of  the  roots  of  the  equation 

x'-\-l.bx^-^0.3x  —  46  =  0. 
If  we  substitute  3  for  x,  the  result  will  be  — 4. 6,  a  nega- 
tive quantity.     If  we  substitute  4  for  x,  the  result  will  be 
43.2,  a  positive  quantity.     Therefore,  the  first  figure  of  the 
root  sought  must  be  3. 

2.  Find  the  first  figure  of  one  of  the  roots  of  the  equation 

xi-j.  3r'+2x-^-f  6x—  148  =  0. 
Putting  2  for  x,  the  result  is  — 88,  and  putting  3  for  z, 
we  get  50,  .*.  the  first  figure  of  the  root  sought  is  2. 

3.  Find  the  first  figure  of  one  of  the  roots  of  the  equation 

a^— 17x"+54x—  350  =  0. 

In  this  example,  the  two  consecutive  numbers  between 
which  there  is  a  root,  are  10  and  20,  therefore,  the  first 
figure  of  the  root  sought  is  1  in  the  ten's  place. 

(264.)  By  actual  multiplication,  we  find 

(x  —  ai)(x — a.)  =  x2         ^Vx-f-cifl.a, 

fll  ^        +^102  i 

(x — a))(x  —  ao)(x  —  a,i)=r'  — a..  >  x-+^i"^  /  ^  —  fliOaflsj 
—  03  7      H-OoCa^ 


GENERAL    PROPERTIES    OF    EQUATIONS.  341 

{x—ai){x—a.>){x—a-3){x—m)=x'  _C3  /  ^' 

—  04   ) 


-j-OiflaX 

-i-aias) 

OiOoO^  -\ 

+  aia4( 

>  a-  + 01020304, 

+  02^3^ 

—  010304  i 

-j-fl-iOiX 

—  020304  ^ 

+  0304/ 

,  &c., 

&c., 

&c. 

By  carefully  examining  the  above  results,  we  discover 
the  following  properties  : 

(265.)   The  coefficient  of  x  in  the  first  term  is  always  1. 

The  coefficicjU  of  the  second  term^  is  the  sum  of  all  the 
roots  with  their  signs  changed. 

The  coefficient  of  the  third  tenrf,  is  the  sum  of  all  the 
products  of  the  roots  taken  two  at  a  time. 

The  coefficient  of  the  fourth  term,  is  the  sum  of  all  the 
products  of  the  roots  with  their  signs  changed,  taken  three 
at  a  time. 

And  so  on  for  the  succeeding  terms,  until  we  reach  the 
last  term,  which  is  independarit  of  x,  and  is  equal  to  the 
continued  product  of  all  the  roots,  ivifh  their  signs  changed. 

(266.)  The  general  form  of  an  imaginary  or  impossible 
root  of  an  equation  is  o-f-  ^ — b. 

The  only  factor  which  will  render  o-f-  ■/—  6  rational,  is 
a  —  ^—h. 

We  have  just  seen,  that  the  last  term  of  our  general 
equation 


342 


GENERAL  PROPERTIES  OF  EQUATIONS. 


is  composed  of  the  continued  product  of  all  its  roots. 

Hence,  if  a-j-  ^ —  6  is  a  root  of  this  equation,  then  also 
will  a  —  ^ —  6  be  a  root,  unless  Jl,^  is  imaginary. 

In  the  same  way,  if  a'  -f  ^ — b'  is  a  root,  then  will  ai —  v^ — h' 
be  a  root,  and  so  for  other  imaginary  roots.  From  this  we 
infer  the  following  properties  : 

(267.)  Every  equation  has  an  even  number  of  impossible 
rootsy  or  else  none  at  all . 

Jin  equation  of  an  even  degree  may  have  all  its  roots  im- 
possible; but  if  they  are  not  all  impossible,  two  of  them  at 
least  are  possible. 

If  all  the  roots  of  an  equation  are  impossible,  then  what- 
ever values  are  substituted  for  x  in  that  equation,  the  results 
will  always  be  affected  loith  the  same  signs. 

An  equation  of  an  odd  degree  has  at  least  one  real  root. 

(268.)  If  we  divide  both  members  of  the  identical  equa- 
tion 

(x  —  a])(x- 
by  x",  we  shall  obtain 


2X"-\...^,,^,x+A=? 
fl,.)(x  —  a.) (x— a„)  ) 


-    ,  j?i   ,   ^3         •/?«— 1   ,  Jin 
XX-  X"-*        X" 


(-?)('-'7)('-?)--(-?)- 

Taking  the  lognritiims  of  both  members,  we  find 


GENERAL  PROPERTIES  OF  EQUATIONS. 


343 


+  loo-. 


+  Iog.(l-f)V(A) 


If  we  actually  take  the  logarithm  of  the  left-hand  mem- 
ber of  (A),  by  formula  (C),  Art.  237,  where 

X  X2 X"-^  X^ 

is  put  for  7J,  we  shall  obtain 

By  taking  the  logarithms  of  the  terms  of  the  right-hand 
member  of  (A),  we  get 

—  (oi-f-ao+as «-.)r 

-^(«l+«^+«^3 «.t)^/ 

-^(«?+«.^+«? <)-2 

&c.,  &c. 


fli-f-aa+fla- 

...a„  =  — ^,, 

a'.+al+al. 

...a==^=-2A>, 

a',+al+al. 

. .  .G,=j  =  —  yi?+3^j^o— 3^3, 

a\+al+al. 

...<  = 

344  GENERAL  PROPERTIES  OF  EQUATIONS. 

By  equating  the  coefficients  of  the  like  powers  of  x,  in 
(B)  and  (C),  we  find  the  following  interesting  properties  : 


(D) 

^5-4^-:^,  -|-4A^3+2^S— 4^4, ' 
&c.,  &c. 

(269.)  These  relations  make  known  the  sum  of  the  7?ith 
powers  of  all  the  roots  of  an  equation  in  terms  of  its  coeffi- 
cients. 

(270.)  If  we  suppose  the  general  equation  is  deprived  of 
its  second  term,  or  which  amounts  to  the  same  thing,  if  we 
suppose  ^1  =  0.  the  above  results  of  (D)  will  become 

a,-\-a,-{-a^....an=0,  ^ 

c?+a^+ai....a-;=-2^2,  / 

aJ  +  a3  +  a^...a3z=-3^3,         )>  (E) 

&c. 


—  3^3,  \ 

&c.  ) 


TRANSFORMATIONS  OF  EQUATIONS. 

(271.)  We  will  resume  our  general  equation 
a;»+^ix"-'+^oa:«-2 -f  A_ix+A  =  0.  (1) 

If  in  this  equation  we  suppose  x  =  u-\-x',  m  being  a  new 
unknown  quantity,  and  x'  an  indeterminate  quantity,  we 
shall  have 

{u-\-x'Y-\-M^+^'Y-'+Mu+x'Y-'  I       /2^ 


GENERAL    PROPERTIES    OF    EQUATIONS.      ^  345 

which,  when  expamled  by  the  Binomial  Theorem^  becomes 


Ax'"-'/ 
+^W  >  =  0.  (3) 


+  ■ 


Now,  since  x'  is  wholly  arbitrary,  we  are  able  to  give  it 
such  a  value  as  to  satisfy  this    condition  7?x'-f-^i  =  0; 

n 

which  is  done  by  making  x'  = ^. 

n 

This  value  of  x'  substituted  in  (3),  will  give  an  equation 
of  the  following  form  : 

W''4-jBotf"-2_^5oM"-3 Bn-iU-\-Bn  =  0^  (4) 

which -is  deprived  of  its  second  term. 

(272.)  Hence,  to  cause  the  second  term  of  an  equation  to 
disappear y  we  must  replace  the  unknown  by  a  new  unknown 
augmented  by  the  coefficient  of  the  second  term  with  its 
sign  changed,  ajid  divided  by  the  number  denoting  the  degree 
of  the  equation. 

•  EXAMPLES. 

1.  Transform  the  quadratic  equation 

into  a  new  equation  wanting  its  second  term. 

A' 
Assume  x^w —      ,  and  it  will  become 

thisj  when  reduced,  becomes 

44 


346  ,    GENERAL   PROPERTIES    OF   EQUATIONS. 


-•^-(f-^i=='' 


or,  by  transposing, 
4 


and,  .  =  ._:|^  =  -|l±V^fZ7, 

The  same  result  as  was  obtained  by  the  direct  solution 
of  the  above  equation  under  Art.  151,  formula  (D). 

2.  Transform  the  cubic  equation 

a;-'  -j-  ^,x»  +  Ji^x + Jl^  =  0, 
into  a  new  equation,  wanting  its  second  term. 


AssuminjT  x  =  m  — 


^1 


we  get 


which,  when  expanded  and  reduced,  gives 


'here 


5o  =  . 


-}-^„ 


We  might  proceed  in  this  way  for  the  transformation  of 
equations  of  higher  degrees,  but  its  easy  to  see  that  this 
method  would  be  very  lengthy  and  complicated  for  such 
equations,  we  shall  therefore  seek  some  law  by  which  these 
transformations  can  be  made  with  less  labor. 

(273.)  If  in  the  general  equation 

x»  -f-  ^ix**-'  4-  ^2X— '-^ 4-  An-xX  -\-An^Q  =  X. 


GENERAL    PROPERTIES    OF    EQUATIONS. 


347 


we  substitute  x'  -\-u  for  a;,  and  imitate  the  operations  of 
Art.  271,  we  shall  have 


+.^,x'»-2         +  (n  —  2)^2x'"-3 1 


+n. 


n  — 1 


=  0. 


w'+...+w« 


If  in  the  above  transformation,  we  put  X'  for  the  coefficient 
of  M°5  or  which  is  the  same  thing,  for  the  sum  of  the  terras  in- 
dependent of  tc.     Also,  put  X"  for  the  coefficient  of  w,  and 

X'"  .  X"" 

— -  for  the  coefficient  of  m^,  — — -  for  the  coefficient  of  w^, 
^  2i.o 

and  so  on,  we  shall  have 

X=  x"  +  ^ia;'-i  + Aia:'-^ ^„_ix-f  A, 

r=a;"'-f-^:x'"-'  +  A2'»-=^ A_ix'+A, 

J^"=nx"»-»-f  (n  — l)^ix'"~^+(n— 2)^2a:"'-l . . .  A-i, 

:X"'=  7i(7i— l)x"»-2-f-  (n  —  1)  (n  —  2).^,x"'-^+ 

Jt""=;2(n— l)(n— 2)x"'-3+(n— l)(n— 2)(n— S).'?,!'"-'.. 
&c.,  &c. 


348  GENERAL  PROPERTIES  OF  EQUATIONS. 

If  we  examine  the  above  expressions,  we  shall  discover 
the  following  law  : 

X'  is  derived  from  the  general  equation  X,  by  simply 
changing  x  into  x'. 

X"  is  derived  from  X'  by  multiplying  each  of  the  terms 
of  X'  by  the  exponent  of  x'  in  that  term,  and  diminishing 
this  exponent  by  a  unit. 

X'"  is  derived  from  X"  in  the  same  manner  as  X"  was 
derived  from  X'. 

And,  in  general,  a  coefficient  of  any  rank,  in  the  above 
transformed  equation,  is  formed  by  means  of  the  preceding, 
by  multiplying  each  term  of  the  preceding  by  its  exponent, 
and  dividing  the  product  by  the  number  of  coefficients  which 
precedes  the  terms  sought,  and  diminishing  the  exponent  by 
a  unit. 

(274.)  The  polynomial  X"  is  called  the^rs^  derivedpoly- 
nomial  of  X'. 

The  polynomial  of  X'"  is  called  the  second  derivedpoly- 
nomial  of  X' ;  and  so  on  for  the  succeeding  polynomials. 

(275.)  We  will  add  a  few  examples  to  illustrate  the  above 
law. 

1.  Transform  the  equation 

X '  —  12x3 -|- 17x2  —  9x-}- 7  =  0, 
into  an  equation  wanting  its  second  term. 

12 

By  Art.  272,  we  must  substitute  w+—  =m-}-3,  or  3  -[- w 

for  x;  this  transformed  by  Art.  273,  will  be  of  this  form  : 
X'JrX"u^'^u'^^'^.u^+u*  =  0. 

Now,  by  the  above  law,  we  find 


GENERAL  rUOrEUTlES  OF  EQUATIONS.        349 

X'  =  (3)'— 12(3)''+17(3)2— 9(3)'+7  =  —  110, 
:^"=4(3)^'  — 36(3)-+34(3)'— 9         =—123, 

-2-=6(3)-'-36(3y+17  =-   37, 

|^==4(3).-12  =  0.      . 

Hence,  our  transformed  equation  is 

w'  —  37m3  —  123;/.  —  110  =  0. 

2.  Transform 

x^  —  I0a:^+7a;3+4x  —  9  =  0, 
into  a  new  equation  wanting  its  second  term. 

Proceeding  as  above,  we  find 

X'  =  {2Y—  10(2;-'  +7(2)^+4(2)^  —  9  =—   73, 
J^"  =  5(2)^  — 40(2)'+21(2)--|-4  =—152, 

^=10(2)^  — 60(2)H21(2)^  =—118, 

^^'=10(2)-^ -40(2)' +  7  =-   33, 

Y'"" 

Hence,  our  transformed  equation  is 

u'  —  33m'*  —  1 18?^^  —  152m  —  73  =  0. 

3.  Transform 

3x-^+15x^+25x  —  3  =  0, 
into  an  equation  wanting  its  second  term. 

Dividing  each  term  by  3,  in  order  to  make  it  agree  with 
the  general  equation,  we  get 

ar^_j_5ar2-J-±:x  — 1=0. 


350 


GENERAL    PROPERTIES    OF    EQUATIONS. 


Now,  in  order  to  make  the  second  term  disappear,  we 
5 
must,  by  Art.  272,  substitute  — ^-\-u  for  x. 
o 


Hence, 


--  (-1) +^  (-B +f  H)- 


25 
3 


_  152 

""  27' 

=  0, 

=  0. 


Hence,  the  transformed  sought,  is 
,       152 


27 


=  0. 


In  this  example,  the  third  term  vanished  at  the  same  time 
as  the  second. 

4.  Transform 

4x3  — 5x'^-f-7a:— 9  =  0, 

into  a  new  equation,  of  which  the  roots  shall  exceed  by  a 
unit,  each  of  the  corresponding  roots  of  the  given  equation. 

We  must  assume  u  =  x-j-1  or  x=u  —  1,  which  gives 

X'=4:  (—1)3  —  5  (— l)-'-j-7(— 1)'  — 9  =  — 25, 
X"=12(— IV— 10(— l)'-f-7  =      29, 

X'" 


l^=12(-l)'-5 


=  —17, 


2.3 


=  4. 


Hence,  the  transformed  equation  is 

4^3— 17tt-'+29u  — 25  — 0. 


(2) 


GENERAL    PROPERTIES    OF    EQUATIONS.  351 

(276.)  The  derived  polynomials  possess  some  remarkable 
properties,  which  we  will  develop. 


x"+A,x^~'+A.^"-^+ +  A_;X+^,  =  0,     (1) 

have  uij  02,  a3, a„_i,  a„,  for  its  7i  roots,  we  shall 

then  have  by  Art.  257,  the  identical  equation 

X-  +  Ax^'  + +  A-isc  4-  A  = 

(x—  ai){x  —  ao) {x  —  c„_i)  {x —  a^). 

In  (2)  change  x  into  x  -\-  v,  and  it  will  become 

{x-\-uY-{-A{x-huY~' ......  ..l.^i{x-\-u)+.(l„  =  )      ,3. 

[u-^{x-a,)\[u-\-ix-a,)] [u -^  {x -  a,.)].  \      ^   ^ 

The  left-hand  member  of  (3),  by  Art.  273,  is 

X  +  X'u-^^^c^-i- w.  (4) 

If  we  should  actually  perform  the  multiplication  of  the 
factors  of  the  right-hand  member,  we  should  find,  by  pay- 
ing attention  to  the  properties  under  Art.  265,  that  the  part 
independent  of  u  is  equal  to 

(x  —  ai){x — aj){x  —  a,;) {x  —  a„_i)(a;  —  a„).     (5) 

The  coefficient  of  u  will  equal  the  sum  of  the  products 

of  all  the  terms  x  —  r/j,  a:  —  ao,  x  —  03, taken  n  —  1 

at  a  time. 

The  coefficient  of  u-  will  equal  the  sum  of  the  products 
of  the  same  terms  taken  n  — 2  at  a  time. 

Hence,  by  equating  the  coefficients  of  the  like  powers 
of  u,  in  the  two  numbers  of  (3),  we  have 


352 


GENERAL  PROPERTIES  OF  EQUATIONS. 


X:=  {x—ai){x — ao)(x — as) . . .  .(x — a„_i)(x — a,,) 

x—ai     X — a-2     X — 03  x — a,j 

X"  =  X  _^  X |_ 

(x— ai)(x— Co)     (x— fli)(x— 03) 


&c. 


(x— a^i)(a:-a„) 
&c. 


(A) 


EQUATIONS    HAVING    EQUAL    ROOTS. 

(277.)  Let  X  denote  the  first  member  of  the  equation 

x^+^iX^-'-f^ax"-' +  A-ia:4- A  =  0,  (1) 

and  suppose  m  factors  equal  to  x — a,  7n'  factors  equal  to 
X  —  b,  m"  factors  equal  to  x  —  c,  &c. ;  also,  that  it  contains 
the  simple  factors  x—p,x—q,x—r,  &c.,  then  we  shall  have 

x=  {x-ar{x-bnx-cr . . . .  ? 

{x-p){x-q)ix-r)...,=  0.    S 

Calling  X'  the  first  derived  of  X,  we  shall,  by  (A),  Art. 
260,  have 

x=i^+t:^+"^+....^+-^+^+..(3) 

x—a     X — b     x  —  c  X — p     X — q     x — r 

Hence,  the  greatest  common  divisor  of  Xand  X'  is 

D  =  {x  —  a)'^\x—b)""-\x-c)  "-'.  (4) 

(278.)  From  this  wc  conclude,  that  when  the  equation 
X  =  0  has  no  equal  roots,  then  the  polynomials  have  no 
common  measure. 

(279.)  If  the  greatest  common  divisor  D,  equation  (4),  is 
of  the  first  degree,  and  equal  to  x — ^  =  0,  we  conclude  that 
equationX  =  0  has  two  roots  equal  to  k.  And  in  general,  if 


GENERAL  PROrEllTIES  CV  EQUATTONS.  353 

it  is  of  the  form  {x — h)"  =  0,  then  the  equation  has  n-f-1 
roots  equal  to  h. 

When  it  is  of  the  form  >r--]-»/':/i.r-|-yio  =  0,  we  must  find 
the  two  vahies  of  a  by  quadraties,  which  we  will  suppose  to 
be  Ic  and  k',  so  that  the  eijuation  will  have  two  roots  ^/c, 
and  two  more  =k'. 

EXAxMPLES. 

1.   Has  the  equation 

2x^  —  12x='+l9a;-— 6a:+9  =  0 
any  equal  roots,  and  if  so,  what  are  they  1. 

X  =  2a- « —  12j:^+  ]  9x-  —  6x+9, 
X'=8x3  — 3Cx'-+3Sx  —6. 
Now,  by  the  method  of  Art.  50,  we  find  the  greatest  com- 
mon measure  of  X  and  A''  to  be 

D=zx-3. 
Therefore,  the  above  equation  has  two  roots  equal  to  3. 
Dividing  its  first  member  by 

{x  —  3Y  =  x^  —  6x-i-9,      . 
we  find 


2x-  —  12a:'+  1 9.r-  —  6x+9 
2x=  — 12r'+lSx- 


6a:-|-9. 


'2a;-H-l. 


X-'  — 6i--f  9 
x3_  6x4-9 

0. 
The  two  roots  of  2x--|-l  ^0  arex^^zb"^ — I. 
Hence,  the  four  roots  of  the  above  equation  are 

3,3,  +  v/=T,  _v/=T. 
2.   Find  the  equal  roots  of 

x^  —  2x4+ 3x''  _  7j;:_|_8x  —  3  =:  0, 
if  it  has  any. 

45 


354 


GENERAL    PROPERTIES    OF    EQrATIO.No. 


X-- 

X'-- 


a-5_2^'_)_ 


+  8X-3, 


~  —  Sx'-\-9--'—Ux  -f-8. 

Seeking  the  greatest  common  divisor  of  X  and  X',  we  find 
D  =  x^  —  2x-{-  I  ={x  —  ly, 

hence,  there  are  3  roots  equal  to  1. 
If  we  divide  the  value  of  X  by 

{x—lY  =  x^  —  3x--\-3x—l 
we  shall  obtain  the  quotient  .r*  -f-  x-|-3. 


The  two  roots  ofx-  -|-3^H-3  =  0,  are  x  ■■ 
hence  the  five  roots  of 

x^  —  2x4  -f  3a;:i  _  7a;2  _|_  Sa:  — 


:±i^— 11 


1,  ],  i,-^4-;v/: 


11. 


3  =  0, 
■^v^— 11. 


3.  Find  the  roots  of 

x-i  +  5x^  4-  6.T^  —  6x^  —  15x3  —  3x2  -f  8x  4-  4  =  0. 

Proceeding  as  in  the  last  example,  we  find 
X=   x^+    5x«4-   6x^—   6x^—15x3  — 3x=+8x  +  4, 
X'  =  7x«  +  30x^  4-  ^0^'  —  24x3  _  45^,0  _  g^  _|.  g^ 
D—    x-«  +  3x3   -j-     X-—    3x  — 2. 

Isow,  since  the  greatest  common  divisor  D,  surpasses  the 
second  degree,  we  cannot  immediately  resolve  it. 

If  we  apply  the  same  process  to  D,  as  we  have  done  to 
X,  we  shall  find 

D=    x' 4-3x3-}-   3,2  _  3a,  _  2, 
D'  =  4x^  4-  92;2  -f  2x  —  3  =  first  derived  of  D, 
D"  =   X  -j-  1  =  greatest  common  divisor  of  D  and  D'. 
Hence,  D  has  two  roots  equal  to  —  1 .     Dividing  it  by 
(^+l?  =  ^^+2x+l, 
we  obtain  the  quotient  x'--\-x  —  2, 
which  equated  to  zero,  gives  x=  1,  or  x  =  —  2. 


GENERAL    PROPERTIES    OF    EQUATIONS.  355 

Therefore,  D=  {x-]-iy{x —  l){x  ^2), 

and  consequently,    A'=  (x  +  ^Yi^  —  l)-(-^"  -\-  ~) ' 
and  the  equation  has  three  roots,  ^=^  —  1  ;  two  roots,  =;  1  ; 

:  — 2. 

RECURRING    EQUATIONS. 

(280.)  v/i  recurring  equation  is  one  which  remains  the 
same    when   -  is  substituted  for  ar. 

X 

All  recurring  equations  are  of  this  form  : 

z"  +  ^ix"-'  +  .V"—' +  Ji'iX-  +  ^ix  +  1  =  0,  (1) 

where  the  coefficients  of  the  terms  equi-distant  from  the 
extremes  are  equal,  because,  if  for  x  we  substitute  -,    the 
above  equation  will  become 

i+r^'-  +i?i ^^^j^i=o,    (2) 

which,  when    cleared  of   fractions  by  multiplying  by  x", 
becomes 

l-\-A,x-^Jl^-^ 4-^ox"---}-^iX''-'+x''=  0,      (3) 

which  is  just  the  same  as  equation  (1),  only  the  terms  are 
taken  in  a  reverse  order. 

From  the  above  definition  of  a  recurring  equation,  we 

know,  that  if  ax  is  one  of  the   roots,  then  W'ill  —   also  be 

a  root  of  this  equation. 

Hence,  recurring  equations  are  sometimes  called  recipro- 
cal equations. 

(281  )  A  recurring  equation  of  an  odd  degree  can  in 
general  be  represented  by 


35G 


GENERAL  PllOrElll  lES  OF  EQ^ATIO^-S. 


a:--+i^i,j'"db.  ?.a-^-'*-'d=  . . .  .±J2rx'-±:.hx±l  =0.       (4) 
Now,  if  the  corresponding  coefficients  have  the  same  sign, 
a:=— Iwill  satisfy  (4),  but  if  the  corresponding  coeffi- 
cients haA'e  contrary  signs,  then  x^=l  will  satisfy  (4). 

(282.)  Heiice^  — 1  or  -\-\  is  ahcays  one  root  of  a  recur 
ring  equation  of  an  odd  degree;  consequently ^  by  Jlrt.  255 
we  know  that  a  recurring  equation  of  an  odd  degree  is  divisi- 
ble by  x-f- l^orhy  x — 1 ;  and  the  quotient  will  be  a  recurring 
equation  of  one  degree  loioer,  and  consequently  of  an  even 
degree. 

(283.)    The  general  form  of  a  recurring  equation  of  an 
even  degree  is 
x2»-|-^iz3"-'-)-^2.r«-2_j_  ^  ^  __j_^,a;2_^^^^j^l  ^  0.         (5) 

This  divided  by  a;",  becomes 

a:"+^,x"^'-f^„a;~-+....-f  4^.4-  "^L^-L^O,       (6) 

which  becomes,  by  bringing  the  terms  of  equal  coefficients 
together, 

^"+x-.+-^'(^  -'+i;!=,)+-^-^(-'-Hj;^,)+&c.=0.  (7) 

If  we  expand  I  .r-| — ;, )  X  I  .r-(--  j)  we  shall   obtain  this 
identical  equation, 
(^'•+1)  X  (.+^)  =."+.+-ip,+.»-.+J^.         (8) 

By  transposing,  we  have 

where  z  =  a:-|-  -. 


GENERAL  PROPERTIES  OF  EQUATIONS.  357 

If  in  formula  (9)  we  suppose  successively 

''  =  1,2,:^,4,5, , 

we  shall  find 


x^-\- 


(A) 


These  values  of  x-f-- ;  x^  4- -^, :  x^ -4- —.:  &c.,  in  terms 
'  X  x~  x^ 

of  z,  being  substituted  in  the  general  recurring  equation  of 

an  even  degree,  will  give  an  equation  in  terms  of  z  of  but 

half  that  degree. 

(284.)  From  Art.  281,  we  know  that  a  recurring  equation 
of  the  degree  2n-[-l,  can  be  immediately  reduced  to  a  re- 
curring equation  of  the  degree  2w,by  dividing  by  x-f-l,  or 
x —  1.  Consequently  a  recurring  equation  of  the  degree 
2?i-f-l  can  be  reduced  to  an  equation  of  the  7ith  degree. 

Suppose,  for  example,  we  wish  to  find  the  five  roots  of 
the  recurring  equation 

x'^—]\x'-\-llx^-\-\1x^—nx-{-l  =  0.  (1) 

Since  this  is  a  recurring  equation  of  an  odd  degree,  and 
the  corresponding  coefficients  have  the  same  sign,  it  follows, 
by  Art.  281,  that  one  of  its  roots  is  —  1.  Dividing  this 
equation  by  x-J-1,  we  obtain  for  a  quotient  this  new  recur- 
ring equation  of  the  fourth  degree. 

x'—\2x^-]-2dx'-—12x-\-l=0.  (2) 


]5S 


GENERAL    PROPERTIES    OF    EQUATIONS. 


Dividing  this  by  x\  and  reducing  the  result  to  the  form 
of  (9),  Art.  283,  we  have 


,.+  i_,2(,+_l)  +  29  =  0. 


(3) 


Substituting,  in  (3),  for  x^  -\ — ^,  a;  -j-  -?  their  values  in 

terms  of  2,  as  given  by  group  (A),  Art.  283,  we  obtain 
s2_  2— 122  +  29=^0,  (4) 

or  z2  — 12z  +  27  =  0.  (5) 

Equation  (5),  solved  by  the  usual  rule  for  quadratics, 
gives 

z  =  9,  or  c  =  3. 

Taking  the  first  value  of  r,  we  have 
1 

X 


9x  = 


Solving  (7)  by  quadratics,  we  find 
9     1     „^ 

Taking  the  second  value  of  z,  we  have 

z  =  x-\ —  =  3,  or  X-  —  3x=:  —  1. 


Equation  (9)  gives 


3     1     ^ 

2^2^^'- 


(6) 
(7) 

(8) 

(9) 
(10) 


Therefore,  the  five  roots  of  the  proposed  equation  are 
9-f-sr77   9— v/77  3-|-v/5   3  —  s/5 
'         2       '       2      '  ~~2      ' 


h 


If  the  numerator  and  denominator  of  the  third  root  be 
each  multiplied  by  9-}-  v/  77,  and  the  numerator  and  de- 
nominator of  the  fifth  root,  be  each  multiplied  by  3-j-  >/  5, 
the  roots  will  assume  the  followincr  form : 


GENERAL    PROPERTIES    OF    EQT'ATIONS.  359 


9-f-^/77         2  34-V/5 


2 


2        '9-|-v/77'       2      '3-f-v/5' 
which  shows  that  the  third  root  is  the  reciprocal  of  the 
secondj  and  the  fifth  is  the  reciprocal  of  tjie  fourth. 

BINOMIAL    EQUATIONS. 

(285.)  Binomial  equations  are  of  this  form  : 

in  which,  if  we  substitute  ax  for  y,  and  divide  the  result  by 
a",  we  shall  obtain 

for  the  general  form  of  binomial  equations. 

(286.)  If  n  is  even,  the  equation  cr" -j-  1  ^  0  ;  or, 
x'*  =  — 1,  gives  for  x  the  impossible  expression  ^ — Ij 
hence  all  the  roots  are  imaginary.  But  the  equation 
X"  —  1  =  0  ;  or  X"  =  1,  gives  a;=yi=-|-],  or  —  1; 
so  that  the  equation  has  two  real  roots,  and  n  —  2  imaginary 
roots. 

(287.)  If  n  is  odd,  the  equation  x'-f- 1^=0  ;  orx"= — 1; 
gives  X  =  V —  1  =  —  1  ;  so  that  there  is  one  real  root  and 
n — 1  imaginary  roots.  But  the  equation  x"  —  1=0;  or 
x"=  1,  gives  x=  V  l  =  -j-  1,  so  that,  as  before,  we  have 
one  real  root  and  n  —  1  imaginary  roots. 

(288.)  If  a  is  one  of  the  imaginary  roots  of  the  binomial 
equation^ 
then  will  x  "=  1  , 

(«■)"  =  «'—  1, 
{a^)"  =  a''-"=  1-=1, 
(a3)"=a3'»=  13=1, 


360  GENERAL    PilOPEUTIKS    OF    E^UATIOKS. 

So  that  a',  a-j  a',  a',  &-C  ,  satisfy  the  equations  =  1, 
when  substituted  for  x.  These  quantities  are  therefore  roots 
of  the  above  equations. 

Hence^  if  a  is  one  of  the  imnginar§  roots  of  the  equation 
rc'*^!,  then  any  j^ower  of  a,  will  also  be  an  imaginary 
root. 

From  this  it  follows,  that  the  roots  a;"=  1,  may  be  rep- 
resented under  an  infinite  variety  of  forms,  each  term  in 
the  following  series  being  a  root. 

1,  fl,  a-,  a3, """'j^ 

a",  a"+^,  a"r2j «^"~S( 

a2",  02"+',  a'^"r2^ a3— 1,  >  (A) 

(289.)  When  n  is  a  prime  number,  the  roots  of  the 
equation  x"=^  1,  are  all  contained  in  either  of  the  expres- 
sions (A),  for  in  each  of  these  series  of  roots  all  the  n  terms 
will  be  different.  But  when  n  is  a  composite  number,  the 
roots  of  the  equation  are  not  all  contained  in  cither  of  the 
series  (A),  for  some  of  them  will  be  the  same  root  under 
different  forms,  for  suppose  7i  =  7;  X  7,  and  let  5'>jo,  then 
the  first  series  of  (A)  is  the  same  as 

1,  «,  a',  o ", fl?',  aP+', a?'+-, a\  a'^', a^-'. 

Now,  since 

therefore  the  terms  1,  «'',  and  a*?,  are  each  equal  to  1,  and 
consequently,  each  must  be  the  same  root  under  different 
forms. 

(290.)  Suppose  we  have  x^=\^  where  p  =  a  prime. 
If  wc  put  xP==y;  then  yf=  1. 


GENERAL    PROPERTIES    OF    EQUATIONS.  361 

Now,  suppose  I  is  a  root  of  i!/'':i^  1,  it  will  follow  from 
Art.  288,  that  the  p  roots  of  3/''=  1  will  be  denoted  by 

1,  5,  h-,  P, bP-K 

Hence,  by  substitution,  we  have 

-X^-1=:0,  (!)• 

xP—b=0,  (2) 

,._3,=  ^-^-^^=0,  (3)1  ^^, 

The  ])  roots  of  the  first  equation  x^ —  1=0,  have  already 
been  found  to  be 

1,  h,b-,  b\ bP-\ 

If  we  make  x  =  c  V  Z),  the  second  equation  of  (B)  will 
become 

xP—b  =  {zr'—\)xh  =  Q; 

therefore  the  roots  of  x^ — 6  =  0,  are  equal  to  the  roots  of 
zi'—  1  =  0  multiplied  by  V  h. 

Hence,  the  p  roots  of  (2)  are 

V  b,  bV  h,  h~  V  b, bP-^V  b. 

Again,  if  we  make  x  =  r  V  b'-,  the  third  equation  will 
become 

xP—b-=  {zP—  1 )  X  6-=  0  ; 
therefore,  the  roots  of  xP — b-=  0,  are  equal  to  the  roots 
of  r''—  1  =  0,  multiplied  by  V  b\ 
Hence,  the  p  roots  of  (3)  are 

V  &-,  h  V  ft'S  b-  Vb--, ft''"'  V  6-. 

Proceeding  in  this  way  m;iy  find  the  following  for  the pp 

roots  of  xPP=  1. 

46 


362 


GENERAL    PROPERTIES    OF    EQUATIONS. 


hb,b', bP~\ 

V6,  h  Vb,  6-  Vb, bP-^  V&, 

V&S  b  V&-,  b^'  Vb% bP-'  V6S 


(C) 


V  bP-\  bVbP-',  b- VbP-\ b^^VbP-K 

(291.)  Again,  suppose  we  have  x^^  —  1  =  0,  where  j9  and 
q  are  both  primes. 

If  we  put  xP  =  y,  we  shall  have  yi  —  1=0. 
Let  the  q  roots  of  this  equation  be 

1,  a,  a-,  a^, a'-', 

or  which,  by  Art  288,  is  the  same  as 

1,  aP,  aP,  cv'P a^t-^^P^ 

then  by  substitution,  we  find 

a;P— 1=0,  (]) 

,  a;P  —  aP  =  0,  (2) 

l^p  — «->=0,  (3)' 


xP—y 


I 


(D) 


xP  —  «(«-')/^=0.  {q) 

We  will  denote  the  values  of  x  in  a-'' —  1  =  0,  by 

1,&,  6-',  6', IP-K 

If  we  make  x  =  or,,  equation  (2),  of  (D),  will  ber,ome 
xP  —  aP  =  {zi  —  \)bP  =  Q; 

therefore  the  roots  of  a^ —  uP  =  0,  arc  equal  to  the  roots  of 
zP  —  1  =  0  multiplied  by  a.  And  in  a  similar  way  we  dis- 
cover that  the  roots  of  xp  —  q'-p  =  0,  are  equal  to  the  roots 
of  o-P —  1  =:  0  multiplied  by  a-,  and  so  for  the  other  equa- 
tions of  (D). 

Hence,  the  pq  roots  of  x^?  —  1  =  0,  are 


GENERAL  PROPERTIES  OF  EQUATIONS.  363 


l,^i^i^ ^^-s 

a,  a6,  ab'j  ab^, afc^"', 

a2,  0=6,  a-6-,  a%', a'^bP-\ 


(E) 


a'-',  a? -'6,  ai-'b'^ai-^b-', o'-'ftP^'. 

As  a  particular  case,  suppose  we  wish  the  15  roots  of  the 
equation  x^^ —  1  =  0,  or  x^-^—1  =0. 

In  this  case,  p  =  3,  and  5=5;  we  must  therefore  seek 
the  roots  of  x^  —  1  =  0,  and  x^  —  1  ==  0. 

We  know,  by  Art.  287,  that  a- 1=  1  will  satisfy  each  of 
the  above  equations  ;  hence  they  are,  by  Art.  255,  both  di- 
visible by  X  —  1.     If  we  effect  the  division,  we  shall  have 

a;2-|-x-]_l=0,  and  x*-\-x^-{-x--j-x-{-l  =  0, 

for  the  results;    the  first  of  these,  x"-|-x-|-l  =  0,  being 
solved  by  quadratics,  gives 

a:=  —  i  +  iN/"^,  or  X  =  —  1  —  iv/^^. 

The  other  equation,  x^-j-x^-j--"^""!"^"!"!  =0)  is  a  recur- 
ring equation.     Dividing  it  by  x^,  ^ve  have 

x=-^x+I+l+?^==o, 

X  X* 


(..+ij+(.+y+i=o. 


Substituting  for  x'--|-— ,  and  a;-j--,  their  values  in  terms 

of  c,  Art.  28",  we  find 

z-'  +  z-l  =  0. 
This,  solved  by  quadratics,  gives 

z  =  -i  +  iv/5,orz  =  — i  — iv/5. 


261  GENERAL  PROPERTIES  OF  EQUATIONS. 

Taking  the  first  value  of ;:,  we  have 
z  =x+i  =  —  i+ 1.  v/5,  or  x- —  (^  v/5  —  i)x  =  —  1, 

X 

which,  solved  by  quadratics,  gives 

:c  =  J  [v/5  —  1+ >/— 10  — 2x75], 
x=  i[v/5  —  1—  v/_lO  — 2v75]. 
Taking  the  second  value  of  z,  we  have 


z  ^  x-\--  =  — 


v/5,orx^+(iv/5  +  i)a:  =  — 1, 


which,  solved  by  quadratics,  gives 


a:  =  — i[v/5  +  l—N/— 10+2^5], 
or  a;=3  — i[^5  +  lH-v/— 10-h2v/5]. 

In  this  case  weVave  for  the  three  roots  of  x^  —  1  =  0, 
the  following  : 

1  =  1,  

62  =  —  ^  —  I  ^/—  3. 

We  have  for  the  five  roots  of  x^ —  1  =  0,  the  following  : 

1  =  1,  

a  =  l[^/5  —  1  4-v/—  10  — 2v/  5], 

a«  =  -  i  [v/5  +  1— v/-10  +  2"75], 

a»  ==  —  I  [v/5  +  l  +  v/— 10  +  2s/5], 

a'  =       ^v/  5— 1— v/— 10— "275], 

Consequently,  the  fifteen  roots  of  a:"' —  1  =  0,  are 

1  =  1, 

a=      ;[v/5  — l+v^— 10  — 2v' 5], 
a-^  =  —  1  [,/5  +  1  —  y—  I0-f-'275], 

a^=  —  i[V5  +  1  +v/^To~+T7^], 

a^=      1  [^5  _l_v/_  10^^75], 


GENERAL    DlOrilUTIES    OF    ECiUATIONS. 


365 


6=_.  [l_v^_3j, 
6a=—  1  [1— v/— 3JX[V5  —  l+v^- 
ba^=       i  [1  —  n/^^3Jx[v5  +  1  —  ^Z- 
ba'=       i[l  —  V^^Jx[^/5  +  1  +v/:_i0_|_2v5], 

62o=— 1  [1  4-v/— 3jx[v/5 

6V  =         Ul  +  ^''  --3  1  X  [  v^O   +   1  —  ^/_10+2^75], 

1  [1  ^  v/Zrr3]  X  [  ,/5  +  1  +  ^/— 10  +  2^5], 
-  -  [1  -|.\/ir3]xrv/5  —  1  — x/— 10— 2^75]. 


10— 2v/oJ, 
T0-f2v/5], 


1  _v/— 10  — 2v/5], 


1  4_x/_iO  — 2v/5j, 


3]x[^/5- 

If  we  extract  the  roots  imlicated,  to  7  places  of  decimals, 

and  reduce  the  results  to  their    simplest  forms,  we  shall 
have 

1=    1,        •         _ 

a=       0.3090170  +  0.95105G5^/— 1,  (3) 

a2  :r^  _  0.8090170-1-0.5877853  V—l,  (6) 

a'  =  —  0.8090170  — 0.5S77S53\/—1,  (6') 

0^=   0.3090170  — 0.95105G5v/—l,  (3') 

6  =  — 0.5000000+0.8660254%/^,  (5) 

ah  —  —  0.9781476  — 0.20791 17  %/—l,  (7') 

a-b  =  —  0.1045285  — 0.9945219%/^,  (4') 

a^b=       0.9135454  — 0.40673G6v/^,  (1') 

a^b=       0.6691306  — 0.7431448  n/^,  (2') 

62  =  _  0.5000000  -  0.8C602"4x/— 1,  (5') 

ab-=        0.6691306  +  0.7431448  n/—1,  (2) 

a"-b'-=       0.9135454-1-0.4067366  v/^,  (1) 

fl-ife^  =  —  0.1045285-1-0.9945219%/^,  (4) 

a'f^  =  —  0.97814764-0.20791 17  v/—l,  (7) 

These  imaginary  roots  are  each  of  this  form. 
And  in  all  cases, 


366 


GENERAL    PROPEKTIES    OF    EQUATIONS, 


For  a  complete  and  lull  discussion  of  the  Binomial  Equa- 
tion^ ac" — 1  =  0,  when  n  is  a  prime,  the  reader  is  referred 
to  the  5th  part.  Vol.  II,  of  Legendre's  Theone  des  JVom- 
hres,  3d  edition^  where  he  will  find  collected  and  demon- 
strated the  many  beautiful  theorems  on  this  subject,  which 
were  first  published  by  M.  Gauss^  in  his  Disquisitiones 
Arithmeticcs. 

(292.)  Before  closing  this  subject,  it  may  not  be  amiss 
to  apprise  the  student,  that  the  solution  of  binomial  equa- 
tions are  most  readily  found  by  the  aid  of  Trigonometrical 
formula. 


GENERAL  SOLUTION  OF  AN  EQUATION  OF  THE  THIRD 
DEGREE 

(293.)  We  have  seen.  Art.  272,  that  an  equation  of  the 
third  degree  may  be  put  under  this  form  : 

x^^A^x-{-Jl^=.0.  (1) 

If  we  assume  x=^y-\-z^  (2) 

we  shall  find  x' =  {y ^  zf  =  y^ -\- z^ -^'iyz^y -[•  z) 


:^-2>yz.x  —  y^-z^  = 

0.                (3) 

If  we  equat 
we  shall  find 

e  the  coefficients  of  (3) 

A,  =  —  3yz, 
A  =  —  y^  —  z\ 

with  those  of  (1), 

(4) 
(5) 

Which  give 

(6)       ' 

y^^z^  =  —  A,. 

(7) 

Cubing  (6), 

we  obtain 

^                               0-7 

(8) 

Squaring  (7),  we  get  /  +  2/z'-f  c«=  ^^       (9) 
Subtracting  four  times  (8)  from  (9),  and  wc  have 


GENERAL  PROPERTIES  OF  EQUATIONS.  367 

Extracting  the  square  root  of  (10),  we  find 


2,3_,3_y^^3+4^?.  (11) 

By  adding  and  subtracting  half  of  (11)  to  and  from  the  half 
of  (7),  we  find 


2+^^    4   +27' 
2        ^     4   ^  27' 


(12) 


(13) 


Hence, 
x  =  y-\-z  = 


If  we  assume 


(14) 


2        *^     4    '    27 
the  above  value  of  x  will  become 

x=^m-\-n.  (15) 

Now,  to  obtain  the  other  two  roots,  we  will  depress  the 
equation 

by  dividing  it  by  r —  {ni-f-n).     See  Art.  255. 


368  GENERAL  PROPEr.TIES  OF  EQUATIONS. 

OPERATION. 

\x''-\-{m-{-n)x-\-  (m  +  n)2-|-^i. 

(m-^n)x''-\-^iX 
{m-\-n)x^ —  {m-^nyx 

[{m-\-ny-\-Jl,]x-\-^2 
[{m-\-ny-\-Ai]x—{m-\-ny—{m-\-n)Ai 

{m-\-?iy-\-{m-\-7i)^i+^.. 

As  it  regards  this  remainder,  we  see  that  since  m-^n  is  a 
root  of  equation  (l)5it  will  be  satisfied  by  substituting  m-j-w 
for  X-  making  this  substitution  in  (1),  we  find 
(m  +  n)3 -(- (m -f- 7^)^i -f- /ia  =  0, 
which  proves  our  remainder  to  vanish. 

Hence,  the  true  value  of  the  depressed  equation  is 

x''-\-{m-\-n)x-\-{vi-\-7iy-\-Jli  =  0.  (16) 

This,  solved  by  quadratics,  gives 


—  {m-{-7,)±  v/— 3  (7/1  +  ny  —  4^1 


(17) 


So  that  equations  (15)  and  (17)  give  the  three  roots  of 
equation  (1 ). 

The  two  roots  contained  in  (17)  may  bo  found  from  (15), 
as  follows  :  Comparing  equation  (14)  with  (12)  and  (13), 
we  find  3/»  =  m',  z^  =  n'' ;  therefore,  by  Art.  288,  we  have 


y  =  am,    >  and         c  =  o;?,    > 

y  =  a'm,  }  z  —  «^72,  } 

where  1,  a,  a',  are  the  three  cube  roots  of  1  ;  that  is, 


GENERAL    PRCPERTILIS    OF    KqrATIONS.  369 

«  =  —  !  4-  i  v/:zi3 .  a-=  —  !  — :  ^—  3. 

See  example  under  Art.  l:  '. 

The  only  \vay  in  which  we  can  combine  the  above  six 
values  of  y  and  c,  so  that  at  the  same  time  their  product 

shall  equal  — '-—,  equation  (6),  is  as  follows  : 

x=:m  +  n,  giving  the  root  of  equation  (15),    "^ 

x  =  a7n-\-a^n. )    ...  „  ,,^,    \     as.^ 

>e[ivino;  tlie  roots  01  equation  (1/),  (      ^      ■' 

The  roots  given  by  (17)  may  be  simplified  as  follows  : 

Since  y  =  m  and  c  =  ?7,  we  have  yz  --^  mn.  Comparing 
this  with  (6),  we  find  ^^i  =  —  3m«,  this  value  of  c^^i,  sub- 
stituted in  (17),  gives 

m  -\-  n   ,   7)1  —  71   /  — r-  "^ 
^  =  --^  +  -^—^-37 

,  C     (19) 

X  = ■ V J.  1 


Collecting  in  one  point  of  view,  the  roots  of  the  equa- 
tion r'  -(-  jTiX  +  JI2  =  0,  we  have 


x=zzm-f-n,  (1)" 

.  =  -'^  +  iLp^=:^,         (2)1 


(B) 


wnere  m  and  71  are  given  by  equations  (14). 

We  will  now  see  what  conditions  must  be  fulfilled,  in 
order  that  one  or  all  of  the  roots  may  be  real. 


47 


370  GENERAL    PROPERTIES    OF    EQUATIONS. 

CASE      1. 

In  this  case,  the  values  of  m  and  7i  are  real,  and  each 
equal  to  \/ — ',  and  the  values  of  x,  given  by  (B),  are 


^V-i\        (1) 


-V=f.  (3) 


CASE    II. 


--(tr+(fr>«- 


In  this  case,  the  values  of  m  and  n  are  both  real,  and 
unequal.  Hence,  the  first  root  as  given  by  equation  (B), 
is  real,  whilst  the  other  two  are  imaginary. 

CASE   III. 
...(-.)%  (-J^o. 

In  this  case,  since  I  -^  1    is  positive  for  all  values  of  j?2. 

it  follows  that  ./3i<0.     This  is  called  the  irreducible  case, 
since  m  and  n  are  both  imaginary. 


GENERAL    PROPERTIES    OF    EQUATIONS.  :'71 

Nevertheless,  we  can  prove,  that  in  this  irreducible  case, 
all  the  roots  are  real.     For, 


Then  we  shall  have 

(C) 


^- =(/>+?  ^—l)^ 


See  questions  13  and  14,  Art.  191,  which  give  the  ex- 
panded form  of  m  and  n  as  follows  : 


(p-9^-1)*- 


J      J    _-'       2     —1         2  5     —  -       

Therefore,  we  find 

2   =''  +5:6''   '-*"=■ 

m— !j      ( 1    -I         2.5      -i      ,   ,      )    , — 
-2-=J3P      9-3^P   V  +  &c.|^_l. 

Hence,  the  three  values  of  t,  as  given  by  (B) ,  become 


372 


GENERAL  PRO'.EllTIES  OF  EQIATIONS. 


\l''''''''-i-'h'''''''''+^'-\'^^'S 


l.r. 


(B") 


)       I,  1    -J         2.5      -I  ,  , 

where  Uie  values  of  y;  and  7  are  given  by  (C). 

And  since  p  and  q  are  real  quantities,  it  follows  that  the 
three  rooU;  as  given  by  (B")  are  real. 

GENERAL    SOLUTION    OF    AN    EQUATION     OF    THE    FOURTH 
DEGREE. 


(294.)  Let  the  equation  of  the  fourth  degree  be  put  under 
this  form  : 

x'4-./J,a;^-f,?,r-f.,^,  =  0.  (1) 

If  we  assume  s  =  y-\-z-\-v,  (2) 

we  shall  find     x-  ■=i/'-\-z--\-u--^2{yz-\-yu-\-zu), 
or  x^  -  {y-^Jrz'-\.u')=  2{yz+yu-^zu).       (3) 

By  squaring  (3),  Ave  find 

X*-    2(r'-fz-^-f-wV-H(?/H~'^+'''')'=      I  /4X 

^y'z'-\-y■^u^+z'u^)-{-Syzu{y-j-z+u).  S  ^   ' 

Replacing  y-^z-\-u  by  x,  in  (4),  and  transposing,  we  find 
x*  —  2{r-{-z-+n^  x^  —  8yzu.x  }  _ 

Now,  in  order  that  (5)  and  (1)  may  become  identical,  w< 
must  have 


GENERAL    PROPERTIES    OF    EQUATIONS.  373 

Ji..  =  —Syzu,  V         (A) 

From  these  conditions,  v:e  immediately  deduce 

yv-^+yV+c=i.=  =:^^IZ_!:i\  ^      (B) 

,  o  o      ^; 
^  64 

Now,  by  Art.  265,  we  know  that  the  sum  of  the  three 
roots  of  a  cubic  equation  with  their  signs  changed,  is  equal 
to  the  coefficient  of  the  second  term  of  that  equation  ;  and 
the  sum  of  their  products  taken  two  and  two,  is  equal  to  the 
coefficient  of  the  third  term  ;  also  the  continued  product 
of  the  three  roots  is  equal  to  the  absolute  term. 

Hence,  the  values  of  y^  ~"'5  and  w-,  will  correspond  with 
the  three  roots  of  this  equation  : 

~  2       ~        16  64  ^   ' 


If  we  suppose  t  =  -.  equation  (6)  will  become 
4 

s'  -{-2.1 16-^  +  (  ^  1  —  4^3)5  —  j21=0. 

If  we  denote  the  three  roots  of  this  equation,  as  found  by 

method  explained  under  Art.  293,  by  s',  s",  a'",  we  shall  have 

u=  ztW^'"-  ) 
Now,  in  order  to  find  x,  a  root  of  (1),  we  must  add  the 
values  of  y,  c,  and  «,  observing  tliat  their  signs  are  so  taken 


374 


GENERAL  PROPERTIES  OF  EQUATIONS. 


that  their  continued  product  may  be  afifected  with  a  contrary 
sign  with  A,^  so  as  to  satisfy  the  second  condition  of  (A). 

CASE    I. 


WlienJi<i<Q. 
The  four  values  will  be  as  follows  : 

x=-\-l^s'—\  Vs"  —  ^ Vs"',{ 
x  =  —Ws'-i-Ws"-Ws"': 
x  =  —Ws'  —  Ws"-\-^Vs"'. 


(D) 


CASE    II. 

When  ^2>  0. 
The  four  values  will  be  as  follows  : 

x  =  —l  s/i'  —  i  Vs"  —  ^  Vs'", 
x=  —  lVs'-\-Ws"-\-^s^s"', 
x  =  -\-W^''—Ws"-\-Ws"', 

X  =  -j-  ^  V^*'  ~l~  I  '^^" i  y/s'". 


(D') 


The  method  of  solving  a  cubic  equation  as  given  under 
Art.  293,  is  generally  supposed  to  have  originated  with 
Cardan^  an  Italian  analyst  of  the  16th  century  j  it  is  there- 
fore frequently  referred  to  as  Cardan's  Method.  Montucla, 
in  his  Hisloire  des  Mathcmatiques^  seems  to  have  proved 
that  it  was  also  discovered  about  the  same  time,  independ- 
ently of  each  other,  by  Scipio  Ferreus  and  JVicolas 
Tartalea. 

The  above  method  for  equations  of  the  fourth  degree, 
which  is  a  close  imitation  of  the  method  for  cubic  equations, 
was  first  given  by  Euler,  a  distinguished  analyst. 

As  yet,  analysts  have  not  been  able  to  obtain  the  general 
solution  of  equations  beyond  the  fourth  degree. 


general  properties  of  equations.  375 

Sturm's  theorem 

(295.)  Let  X=0  be  an  algebraic  equation  having  real 
coefficients  ;  we  will  suppose,  also,  that  it  has  no  equal  roots. 
Call  A'l  its  first  derived  polynomial^  found  by  the  method 
of  Art.  273. 

Apply  to  X  and  Ai  the  method  of  finding  the  greatest 
common  measure,  as  explained  under  Art.  50,  with  this  con- 
dition, always  to  change  the  sign  of  the  remainder  at  each 
operation^  and  to  use  this  remainder,  thus  modified,  for  a  di- 
visor in  the  next  operation. 

Designate,  moreover,  by  An,  As,  A4, Ar,  the  succes- 
sive remainders,  taken  with  contrary  signs. 
If  we  denote  the  successive  quotients  by 

?t5  92,  ?3J qr^\, 

we  shall  have  the  followino-  relations  : 


A=A,7,  — Ao,  (1) 

Ai  =  Ao<7,  — A.,  (2) 

A2  -^  A35  —  A4,  (3) 


A.-o==X-i«7._i  — A..         (r-1) 


(A) 


We  shall  necessarily  have  Xr  independent  of  a*,  and  dif- 
ferent from  zero,  (Art.  278.) 

After  having  obtained  the  functions  A,  Ai,  Ao, Ar, 

suppose  we  substitute  in  them  for  x,  two  numbers  p  and  q 
of  any  signs  whatever,  p  being  <  q. 

The  substitution  of  p  will  give  results  either  positive  or 
negative  ;  if  we  only  take  account  of  the  signs,  and  write 
them,  one  after  another  in  a  line,  they  will  give  a  certain 
number  of  variations  and  permanences. 


376  GENERAL    PROPERTIES    OF    EQUATIONS. 

The  substitution  of  q  will  in  like  manner  give  a  succes- 
sion of  signs,  of  a  certain  number  of  variations  and  perma- 
nences. 

Now,  the  Theorem  of  Sturm  consists  in  this  : 

The  difference  between  the  number  of  variations  given 
by  the  first  series  of  signs,  and  the  number  of  variations  given 
by  the  second  series  of  signs,  will  express  exactly  the  number 
of  real  roots  of  the  proposed  equation,  which  are  comprised 
between  p  and  q. 

(296.)  We  shall  now  proceed  to  demonstrate  this  beau- 
tiful theorem, 

I.  Consider  the  function  A'  in  particular,  and  suppose  a, 
is  a  real  root  of  A  =  0.  If  we  substitute  a[  -j-w  for  x,  in  X,  wr 
shall  obtain.  Art.  273,  a  result  of  this  form  : 

^+^'u-{.'^u^  +  f^tc^ w";  (1) 

where  ^  is  w'hat  X  becomes  when   ai  is  put  for  x,  and 

v4',  ^",  ^"', are  the  successive  derived  polynomials 

of  J?,  found  by  the  method  of  Art.  273. 

Now,  by  hypothesis,  ai  is  a  root  of  AT ^  0,  therefore  j?  ^  0, 

and  the  preceding  expression  becomes 

(an         am  \ 

^^'+2''+l:3^'+ -^''r       ^^^ 

We  can  always  take  u  sufficiently  small  to  cause  the  quan- 
tity within  the  parenthesis  of  (2)  to  have  the  same  sign  as 
its  first  term  A' . 

II.  If,  in  the  functions  A,  Ai,  A2, Ar,  ive  substi- 
tute any  quaiitity  a  for  x,  it  cannot  happen  that  two  conse- 
cutive functions  shall  vanish  at  the  same  time. 

For  take  any  three  consecutive  functions  as  An-i,  An,  X.+i- 
Then,  by  conditions  (A),  w^e  have 

X„.,  =  Xnq.-Xn+U  (1) 


GENERAL  PROPERTIES  OF  EQUATIONS.        377 

Now,  if  we  arc  able  to  have  at  tlie  same  time 

X->=0,  (2) 

X.  =  0,  (3) 

we  must  also,  by  comlition  (1),  have 

AVi  =  0.  (4) 

Since  the  relation  (1)  is  general,  it  must  be  true  when  we 
write  n-\-l  for  7i;  hence  we  have 

Xn  =  X,-^i9«+i  —  Xt+J.  (5 ) 

In  (5),  substituting  the  values  of  X„,  X,+i,  as  given  by 
(3)  and  (4),  and  we  obtain 

a;+2  =  0.  (6) 

By  continuing  this  process,  we  should  finally  find 

a;  =  o,  (7) 

which  is  absurd,  since  we  have  already  shown  that  Xf  can- 
not equal  zero. 

III.  The  relation  X„-i=X„qn  —  Xn-\.i,  shows  that  if  a 
function  X^  becomes  0  by  the  substitution  of  x  =  a,  the  two 
functions  X„_i,  X„_|-i,  between  which  it  is  'placed^  have  ne- 
cessarily contrary  signs  for  x  =  a. 

(297.)  Designating  by  k  a  quantity  positive  or  negative, 
but  less  than  each  of  the  the  real  roots  of  the  equations, 
Z  =0, 

.       X,  =  0X  (B) 


X-i-0, 


Conceive  that  the  value  of  a:  is  made  to  increase  continu- 
ously from  X  =  kj  and  that  its  successive  values  are  substi- 
tuted in  the  functions  X,  Xi,  Xq, Xr.     Now,  so  long 

as  the  increasing  values  of  x  are  less  than  each  of  the  roots 
of  equations  (B),  the  signs,  arising  from  their  substitution  in 
48 


378  GENERAL  PROPERTIES  OF  EQUATIONS. 

the  functions  of  X,  Xi,  X^, X,  will  occur  in  the  same 

order ;  for,  in  order  that  the  number  of  the  variations  and  per- 
manences of  signs  should  change,  it  is  necessary  that  some 
one  of  the  above  functions,  as  X„,  should  have  passed  through 
the  stage  in  which  X,^  =  0,  which  cannot  have  happened, . 
since  x  is  supposed  less  than  the  least  value  which  can  satisfy 
either  of  the  equations  (B). 

(298.)  We  will  now  suppose  that  x  has  reached  a  value 
x  =  a,  which  causes  one  of  the  intermediate  functions  X], 

Xo,  Xs, X-i,  to  vanish,  without  causing  x  to  vanish. 

We  will  also  suppose  X^  to  be  the  one  which  vanishes  when 
x  =  a;  then  by  II,  under  Art.  296,  we  know  that  X,-i, 
Xn-^],  cannot  vanish,  and  by  III,  under  the  same  Art.,  we 
also  know  that  X,_i  and  A',4-1  must  have  contrary  signs. 
Now,  if  we  consider  the  sign  of  the  vanishing  term  Xn  to  be 
either  plus  or  minus,  the  three  consecutive  functions  Xi-ij 
X„,  X.-i-i,  can  produce  only  these  two  combinations  of  signs, 

or  _  ±   4-  S 

Either  of  which  gives  one  variation  and  one  permanence. 

We  know,  by  Art  297,  that  the  signs  of  X„_i,  X„-|-i,  will 
not  be  changed  from  x=k  to  x  =  (i,  and  since  we  are  able 
to  take  u  as  small  as  we  please,  it  follows  that  they  will  not 
be  changed  from  a:  =  ato  x  =  a-\-ti. 

Hence\,  the  hypothesis  x  =  a,  introduced  in  the  series  of 

functions  X,  X|,  Xo, ,  ca7i  produce  neither  a  gain  or 

loss  in  the  number  of  variations. 

(299.)  We  will  now  suppose  that  x  =  Oi  causes  X  to  va- 
nish.    Let  [/and  l]\  represent  the  values  of  X and  Xi,  when 

Represent,  as  in  Art.  273,  by  A^  A' ^  A", ,  the  va- 


GENERAL    PROPERTIES    OF    EQUATIONS.  379 

Ities  of  A'  and  its  successive  derived  functions,  when  a:=  a. 

In  the  same  way,  represent  by  ^5,,  ..^/,  ^i", ,  the 

■values  of  X^  and  its  successive  derived  functions. 
By  Art.  273,  we  shall  have 

Since  Ci  is  a  root  of  A'=0,  we  must  have  Jl  =  0. 
Again,  the  values  A'  and  ./5,  each  represents  the  value  of  Xi 
when  a,  is  put  for  a-,  and  since  the  equation  A:=:0  is  sup- 
posed not  to  have  any  equal    roots,  A'  or  its  equal  cannot 
vanish,  therefore  (C)  becomes 
/I" 

2 

(D) 

01  which  the  right-hand  members  will  have  the  same  signs 
as  their  first  terms  Jl'u,  Jl',  if  we  take  u  sufficiently  small. 

Hence^  when  u  is  positive^  U  and  U^  will  have  the  same 
si  on. 

When  u  is  negative,  U  and  C/j  will  have  contrary  signs. 

From  which  it  follows  that  the  functions  X  and  Aj  will 
give  a  variation  for  x  =  ai  —  m,  and  a  permanence  for 
x  =  a,  -(-  w. 

Consequently,  in  the  passage  of  the  continuously  increas- 
ing values  of  xfrom  x  =  ai  —  u  to  x  =  ai-\-  u  a  variation 
will  be  changed  into  a  permanence. 

The  same  results  would  have  place,  if  the  value  x  =  ai, 
which  causes  A  to  vanish,  should  at  the  same  time  cause 

some  one  or  more  of  the  functions  Ai,  A2,  A3, to 

vanish.     (Art.  298). 

Now,  commencing  with  z  =  Ci-j-w,  if  we  suppose  the 


380  GENERAL  PROPERTIES  OF  EQUATIONS. 

value  of  X  to  increase  continuously,  the  number  of  varia- 
tions in  the  series  of  signs  ^vill  remain  the  same,  although 
the  order  of  the  succession  of  the  signs  may  be  changed 
until  we  reach  another  value,  ar=  02,  which  causes  X  to 
vanish,  and  which  is  therefore  a  root  of  X=^0  ;  in  which 
case  a  second  variation  must  be  changed  into  a  permanence  ; 
and  so  on. 

HencCy  the  number  of  variations  lost  when  x  increases 
from  X  =  k  to  X  =  k' ,  must  be  equal  to  the  number  of  real 
roots  of  X=^  0,  comprised  between  k  and  k'. 

APPLICATION    OF    STURM'S    THEOREM. 

(300.)  Before  passing  to  the  application  of  this  theorem, 
we  shall  do  well  to  pay  attention  to  the  following  principles  : 

I.  In  obtaining  the  functions  X,  Xj,  Xo, A',,  we  are, 

by  Art.  53,  at  liberty  to  introduce  or  suppress  any  numeri- 
cal factor,  provided  that  it  is  positive  ;  but  it  is  necessary 
to  pay  particular  attention  to  the  signs,  and  make  only  the 
changes  mentioned  under  Art.  295,  as  the  peculiarities  of 
this  theorem  depend  principally  upon  this  change  of  the 
signs  of  X,  Xi,  X-, . . . .  X;.. 

II.  If  we  simply  wish  to  know  the  total  number  of  real 
roots,  without  fixing  in  any  manner  their  limits,  we  need 
only  substitute  in  the  first  terms  of  X,  X),  X^, . . .  .X^,  the 
values — CO  and  -\- cc. 

EXAMPLES. 

1 .  How  many  real  roots  has  the  equation  8x^ — 6.r — 1=  0  ? 
The  first  derived  of  Sx^  —  6a;  —  1  is  24a:-  —  G,  or  sup- 
pressing the  positive  numerical  factor  6,  it  becomes  4x- —  1. 

Now,  applying  to  8x^  —  6a:  —  1  and  4a;-  —  1  the  method 
of  finding  the  greatest  common  divisor,  we  obtain  — 4a; — 1 
for  the  first  remainder,  changing  its  signs  it  becomes  4a;+l, 


GENERAL  PKOPERTIES  OF  EQUATIONS.  381 

continuing  the  operation  ^vith  4a;'- —  1  and  4z-f  ^j  ""^'^  ^"^^ 

—  3  for  the  remainder,  hence  we  have 

X=Sx'  —  6x  —  l,\ 
X,  =  4x^-1,  ( 

A',=:4xH-l,  (         ^^^ 

^3  =  3.  ) 

Now,  if  for  X  in  the  above  functions  we  substitute  —  co, 
the  signs  of  the  results  will  be 1 \-  giving  3  variations. 

If  we  substitute  +  co,  they  will  be  +  +  +  +  giving  0 
variations. 
Hence, 

If  in  the  same  functions  (A),  we  substitute  the  three 
consecutive  values  x=  —  l,a:=:0,  x  =  l,  we  shall  find  that 

for  x^ —  1  the  signs  are 1 \-  giving  3  variations, 

"    0-  =  0  "  (-  +      "       1         " 

"■   x=l  "  +  4.  4_  -f      "       0         " 

Hence,  two  roots  lie  between  —  1  and  0  ;  and  one  root 
between  0  and  1. 

If  we  substitute  x  =  —  1 5  "^ve  shall  find  -f-  ± (-  giving 

2  variations. 

T^herefore,  one  of  the  negative  roots  lies  between  —  1 
and  —  ^,  and  the  other  between  —  ^  and  0. 

2.  How  many  real  roots  has  x^  —  5x--}- 8x  —  1  z=  0  ? 
In  this  example,  we  find 

A'=x3  — 5x-^-f-8x  — 1, 
X=3r*— lOx  +  S, 
X^  =  2x  —  31, 
Xi  =  — 2295. 

When  X  =  —  00,  we  find j ,  giving  2  variationg, 

«      x  =  -foo,      «       +  +  +-,     «       1         " 


382  GENERAL    PROPERTIES    OF    EQUATIONS. 

Therefore,  the  above  equation  has  but  one  real  root,  and 
consequently,  it  must  have  two  imaginary  roots. 

3.  How  many  real  roots  has  x*  —  2x^  —  7ar^-{- lOx-j- 
lO  =  0  ? 

In  this  example,  we  find 

A"  =  a;^  —  2a;^  —  7ar^ -f- lOx -f  1 0, 
Xi  =  2x^  —  3x^  —  lx-\-5, 
X.2=  17x-  — 23a;  — 45, 
X3=  152a:  — 305, 
X4  =  524785. 

When  X  =  —  00,  we  find  H i [-,  giving  4  variations. 

«     x=+oo,      "       +  +  +  ^-_f,     "       1.       " 

Consequently,  the  roots  are  all  real. 
We  also  find 

-|-  -(-   — \-  giving  2  variations, 

+ + 

+ + 

4-  +  +  +  + 
-+  +-  + 
+-  + 

"     x  =  -3         -f  -    +-  + 

From  which  we  see  that  the  equation  has  two  positive 
roots  between  2  and  3  ;  one  negative  root  between  0  and 
—  1  ;  and  one  negative  root  between  —  2  and  — 3. 

4.  How  many  real  roots  has  2x* — 13x'4-10x — 19=0? 
Here  we  find 

X  =  2x'  —  13x2  _j_  lOx  _  19, 
Xy  =  4x»  -  13x-f  5, 
^2=  13x'^  — 15X+38. 
It  is  not   necessary  to  calculate  Xn  and  X4,  since  the 
two  roots  of  ^2=  13x2 —  l5x-|-38  :=0,  are  imaginary,  for 
(15)^<4  X  13  X  38.     See  Art.  149,  Formula  (B). 


hen 

X  = 

0 

X  = 

1 

x  = 

2 

x  = 

3 

X  ::=:  - 

-1 

X  =  - 

-2 

2 

2 

0 

3 

3 

4 

GENERAL    PROPERTIES    OF    EQUATIONS,  383 

Using  only  the  values  A'',  Xi,  Xo,  we  have 

■whenx  =  — go  -\-  —  +   giving  2  variations, 

«     a;  =  +oo  4-   +   -I-      "       0         " 

Therefore,  the  two  remaining  roots  are  real. 

5.  How  many  real  roots  has  a:^ — 36x^+72x2 — 37a:-j-72  =  01 
Here  we  find 

X  =a:^  — 36x^-1-72x2  — 37a; -f  7  2, 

Xi  =  5x'  —  108x^  +  144x  —  37, 

Xo=  18x3  — 54x--f-37x— 90, 

X-.i  =  1319x^  —  2487x  —  684, 

Z4  =  —  2960933x-f- 34935426, 

Xs  =  - 
whenx  =  —  00  we  have i pH gi'^'i'^g^  variations, 

"       X=:-f-00  "  +  +  +  H "  1  " 

Hence,  the  proposed  equation  has  three  real  roots  and  two 
imaginary  ones. 

.  6.   How  many  real  roots  has  x^-{-M,x-^^2'==-0  1 
In  this  example,  we  find 

X  =x^'-(-.^,x-{-^,, 

Xi  =  3x--(-^:, 

X,  =  — 2.;^ix  — 3A', 

X.,=--  — 4^?— 27^=. 

CASE   I. 

WAen— 4^J— 27^=>0. 

Now,  since  — 27^P,  is  negative  for  all  values  of  .//■_>,  it  is 
necessary  that  ^,<0,  in  order  to  fulfil  the  above  condition. 
Consequently, 

when  x^  —  x  we  have 1 \-  giving  3  variat-oro. 

"      x=  +  oo        "       -I-  4-  -^  4.      "      0         " 


384 


GENERAL    PROPERTIES    OF    EQUATIONS. 


Therefore,  when -4^?— 27^^  v>o  or  4^?4-27^§<0,  then 
the  three  roots  are  real.     See  Case  III,  page  370. 

CASE   II. 

When  —^J^,  —21^l<0. 

This  condition  can  be  fulfilled  for  values  of  ^i  either  posi- 
tive or  negative,  so  that 

when  x  =  —  oo  we  have f-  db  —  giving  2  variations. 

"     a;  =  -f-oo        "        -f--f-=F—      "       1         " 

Therefore,  when  —  4^?  —  27^-:<0,  or  4^?+27^p0, 
then  there  will  be  but  one  real  root,  and  consequently  two 
imaginary  roots.     See  Case  II,  page  370. 

CASE    III. 


When  —  4.^3  _  27^52  =.0. 

In  this  case,  we  know,  by  Art.  279,  that  there  are  two 
equal  roots  which  will  be  given  by  2^ia;-|-3^2=  0.     Hence, 

one  of  the  equal  roots  is  a:  =  —  -—^,  and  the  other  root  muK 


2^i' 


SJlo 


bex  =  — — .     See  Case  I,  page  370. 

7.  How  many  real  roots  has  x^-{-jiiX--\-^2X-\-^3  =  0  1 

T^ere  we  find 

X  =x*+Aa:^  +  .V4-^3, 

^i=4a:»+2^iX+A., 

Xz=z  —  2j?,x-  —  3^20:  —  4^.,, 

Jr4=  16^3(^?  —^Ji■^)^-Al{^A\—\UJ],A..-\-'ilAl^ 


GENERAL    PROFF.KTIES    OF    EQUATIONS.  385 


CASE    I. 
Whe7i  .'2i<0,  S.^fJh—  2./^?  —9^5  >0,  and 

Then, 
when  x  =  —  ex,  we  find  -\ ( [-,  giving  4  variations. 


CASE    II. 

When  16^i3(^5=  —  4.^3)K^::(4^??  —  144^1^3+27^^). 

Then, 
when  r=  —  oo,  we  find  -j 1 ,  giving  3  variations. 

"    x  =  -^oo,      "       +-j ,     "       1         « 

Therefore,  in  this  case,  there  must  be  two  real  roots,  and 
consequently,  two  imaginary  roots. 

When  neither  of  these  conditions  are  fulfilled,  all  the 
roots  are  imaginary, 

GENERAL    METHOD    OF    ELIMINATION    AMONG    EQUATIONS 
ABOVE    THE    FIRST    DEGREE. 

(301.)  Suppose  we  have  two  equations,  each  containing 
X  and  y,  represented  by 

X=0,  (1) 

^,=0,  (2) 

Now,  if  we  seek  the  greatest  common  measure  of  the 
polynomials  .Y  and  Xi,  by  the  method  of  Art.  50,  we  shall 

have 

X=X,q-^r,  (3) 

49 


386  GENEUAI.    PROPEKTIKS    OF    EQUATK  .<;.. 

where  q  is  tlie  nuolicnt  of  X  divicled  by  X-i,  and  r  is  the 
remainder.  Now,  since  by  (^l)Tind  (2),  Jt  and  X\  are  each 
zero,  it  follows  thai  r  as  given  by  (3),  must  also  be  zero. 

(302.)  FrGiu  which  we  conclude^  thai  if  ire  operate  upon 
the  polynomials  X  fmd  X],  hy  the  method  for  finding  the 
greatest  commuiv  measure^  we  shall  have  the  successive  re- 
mainders each  equal  to  zero. 

If  wc  arrange  the  j.olynoinials  with  reference  to  either 
of  the  letters^  before  operating  vpon  them.,  we  shall  iilti- 
mately  find  a  reni(i)ider  independent  of  that  letter.)  when 
the  polynomiiils  have  iw  common  measure.^  which  remainder 
being  'put  eqind  to  zero,  will  give  an  equation  containing 
hut  one  nnknoirn  quiintity. 

When  the  two  polynomials  have  a  common  measure.,  it 
must  be  put  equal  to  zero.,  if  it  contains  both  the  unknown 
quantities.,  then  divide  both  polynomials  by  it.,  and  proceed 
with  the  results  as  in  the  first  case. 

Note. — In  the  operation  of  finding  the  greatest  common 
measure  of  two  polynomials,  it  frequently  becomes  neces- 
sary to  suppress  factors,  as  well  as  to  introduce  new  factors. 
When  this  is  done,  we  must  carefully  examine  whether 
such  ffictors  are  able  to  effect  the  final  result.  If  no 
factors  other  ihan  numerical,  are  either  suppressed  or 
introduced,  then  the  above  method  is  rigidly  correct,  but 
in  other  cases,  the  rule  would  require  some  modification. 

EXAMPLES. 

].  Obtain  from  the  two  equations 

•^=+-r.y+!/'^-i  =  o,  (1) 

x'-j-y'=0,  (2) 

a  single  equation  in  terms  of  y. 


GENERAL    PROPERTIES    OF    EQIATIONS.  387 

Proceeding  by  the  method  of  finding  the  greatest  common 
measure,  Art.  50,  we  have  for  the 

FIRST    OPERATION. 

^  ~ry  lar-j-yx-j-i/*  —  1 


—  yx-  —  y-x  —  y'^-\-y 


'x  —  y. 


x-j-^T/' —  y  =  first  remainder. 

Again,  dividing  X2-\-yx-^y- —  1  by  this  remainder,  we 
find  for  the 

SECOND    OPERATION. 
X^J^yxJ^-y  —  1  I X-|-2^3  _  y 

x-'J^{2y'-y\r 


—{2y^-2y)x-^f—\ 

— (2r-2.v)x-47y«+6y^— 27/- 

4,1/''— 6j^^-|-3i/- — 1  =  second  rem. 

Putting  this  rema  nder,  which  is  independent  of  x,  equal 
to  zero^  we  have  for  the  equation  sought : 

^.v'-6i/'  +  3r-l=0.  (3) 

If  we  were  required  to  find,  from  the  above  two  equations, 
one  single  equation  in  terms  of  x,  we  observe,  that  all  that 
would  be  necessary  would  be  to  change  y  into  x,  in  equation 
(3),  since  x  and  y  can  be  changed  the  one  for  the  other  in 
equations  (1)  and  (2)  without  affecting  their  form. 

2.  Obtain  an  equation  independent  of  y  from  the  two 
equations 


3S8 


GENKSAL  PROPKRTiES  OF  EQCATiONS, 


H22/H32/-4)r+y— 1  ) 


(2) 


Proceeding  with  these  equations  agreeably  to  the  above 
method,  we  find  for  the  first  remainder  the  following: 

Repeating  the  process,  we  find  for  the  second  remainder, 
the  followin  g: 

which  being  put  equal  to  zero,  gives  for  the  equation  sought, 

3.  Obtain   an  equation  independent  of  y  from  the  two 
equations 

3x»y '  +  (3x-  -  3x)y'  —  (2x-^^  x)f  j  ^  ^ 

+  (x  •  —  2x--' +2x  —  3  )2/+x '  —  X— 2  S 
3x2^3  _  2xy'2  —  (2x2  _  x)y-^x^^x  —  3=0.        (2) 
Ans-  X"  — X  —  2=0. 


(1) 


(303.)  When  we  have  three  equations  involving  three 
unknown  quantities.  We  must  first  eliminate  one  of  the  un- 
knowns by  combining  either  of  the  equations  with  the  other 
two  ;  we  shall  thus  obtain  two  new  equations  involving 
only  two  unknown  quantities,  which,  as  we  have  just  shown, 
will  give  a  final  equation  involving  but  one  unknown  quan- 
tity. 


1.  Obtain  an  equation  containing  only  x,  from  the  thret 
equations 


GENERAL  PROPERTIES  OF  EQUATIONS.  389 

x-\-y^  —  a=0.  (1) 

y+z'-h=0.  (2) 

^^^_c=0.  (3) 

Eliminating  z  between  equations  (2)  and  (3)  we  have  the 
following 

OPERATION, 

z^-\-y  —  6  z-\-x-  —  c 

2^-(-(x-  —  c)z 


z  —  {x'-c). 


—{x^—c)z-i-y—b 
—{x"-—c)z-x*-{-2cx-—c^ 

x^  — 2cx'^-\-y-\-  c^ — 6  =  remainder. 

Putting  this  remainder  equal  to  zero,  we  have 

x'—2cx~+y-^c'—b  =  0.  (4) 

Now,  eliminating  y  between  equations  (1)  and  (4),  we 
have  the  following 

OPERATION. 

yi^x  —  a  \y^x'~-2cx''-\-c-  —  b 

j;J-j_(a;i  —  2cx--{-c- — b)y 

1  j/  —  (x'  —  2cx^  -\-  c-  —  b). 

—{x^  —  2cx--\-c'^  —  b)y-\-x—a 

— (x'  —  2cx--(- c-  -  b)y  —  {x*  —  2cx=  -}- c^  —  6)= 

{x*  —  2cx--\-c''' —  b)--\-x —  az=  remainder. 
Expanding  this  remainder,  and    then  equating   it    with 
zero,  we  have 

x«  — 4cx«-f-(6c-2— 26)x'  — (4^  — 4/)c)x^)_ 

-|_x_(_c'-f-6-^— 26c-  — aS~    ' 
By  simply  permutating  these  quantities,  (Art.     85),  we 
have 

/_4ay«-f-(6a=— 2cy— (4a^— 4cay>^ 

_^3/_|_a.  +  c2  — 2ca-  — 6S         '     ^   ^ 


390 


GENERAL    PROPERTIES    OF   EQUATIONS. 


_|_;_j_/,4_(_f,2_2a63  — c! 


0.     (7) 


2.  In  a  similar  manner,  find  three  equations  each  contain- 
ing but  one  unknown  quantity,  from  the  three  equations 
a:2_^^.y_.a  =  0,  (1) 

r-{-yz-b  =  0,  (2) 

::2_{_z2;— c  =  0.  (3) 

thirst,  eliminating  z  between  (2)  and  (3),  we  find 

y^  —  xy'  —  {2b-\-c)y'-\-bxy-^b'  =  0.  (4) 

Secondly,  eliminating  y  between  (1)  and  (4),  we  find 
2x«— (7o-f-36-|-c)x«  +  (9a-^-f-5a6  +  2crc-f  ^•^)a:^; 
—  (5a3 -f- 2a'6 -f  a2c)x-^-l- a^ : 
By  permutating  these  letters,  we  find 


0.(5) 


22' 


—  (56-^-f-26-^c-f  6'a)?/ 
(7c+3a+fe)c«  +  {9c^-\-5ca  -f  2cb  +  a^)z' 

—  {oc'-\-2c-a  -f  c^b)z-'-\-c 
If  a  =  1€,  6  =  17,  and  c  =  18. 
Then  will  the  eight  sets  of  values  be 

rx=:iz  4.173281, 
1.  <  y  ^  ±  4.287098, 
(z  =  ^  0.330363. 
rx  =  ±  2.525516, 
■2.}y=zt  2.969156, 
(z=  ±  3.240579. 
rx  =  ±  0.418924, 

3.  )  y  =  i  3.912240, 
(z=  ±  4.048877. 
r  X  =  ^  4.003756, 

4.  )  7/  =  dr  0.007100, 
(  c  =  =F  4.245971. 


5:S=o,(6) 
i(=o.(7) 


GENERAL  PKOPmiTlES  CF  EQUATIONS.  3!>1 

3.'  Find  three  Li[uations  each  containing  but  one  unknown 
quantity,  from  the  three  equations 

.•  +  1/c  -  a  =  0,  (i) 

J--}-  zx  —  b  =  0,  (2) 

r  +  a-2/  -  c  =  0.  (3) 

Operating  as  in  the    preceding    examples,  we  find  the 

following  results  : 

x^  —  ax*  —  2x'-\-{2a  +  bc)x'  —  {b'^-\-c-—l)x-\-bc  —  a  =  0j 
3/S— 6y'  — 2y3+(26+a0y-— (C-+0-— l)y+cr  — i^.O, 
z^—cz'  —  2z'-{-:^c+ab)z''—^n'+h'—l)z-{-ab  —  c  =  0. 

4.  Find  three  equations  each  containing  but  one  unknown 
quantity,  from  the  three  equations 

ar-]-yz  —  a  =  0,  (1)' 

y'  +  zx-  I  .     0,  (2) 

z-  +  ay  —  c  =  0.  (3) 

'8x^  — 20aa;«-f  (I8a2  — 26c)a:'  ? 

+  {babe  —  7a'' —  b^ —  c')x'-+  (a-  —  bcY 


=  0, 


^^g    ,8]/«  — 20&y«+ (186--  + 2ca)  ^ 


4-  {bbca  —  lb^  -  c3  — a3)3/^4-(ft2_  ca)2 

82«  — 20cz''  +  (l8c'— 2flZ;)  c'  > 

+  (5ca6  — 7c^  — a^  — 6-^)c-+(c-^  — a6y^S~~    ' 

(304.)  When  there  are  foui  equations,  we  must  first 
reduce  the  number  to  three  by  eliminating  any  one  of  the 
unknown  quantities,  and  then  proceed  as  above.  From 
what  has  already  been  done,  it  will  not  be  diflficult  to  know 
how  to  proceed  for  a  greater  number  of  equations,  but  it  is 
obvious  that  in  many  cases  this  general  method  must  be 
very  long  and  laborious,  still  it  is  valuable  on  account  of 
the  certainty  of  the  result. 


392        NUMERICAL    SOLUTION    OF    HIGHER    EQUATIONS. 


CHAPTER  XL 


NUMERICAL  SOLUTION  OF  CUBIC  EQUA- 
TIONS,  AND   EQUATIONS  OF 
SUPERIOR  DEGREES. 

(305.)  Let  ^ix5-j-^2x'+^3X  =  A  (1) 

be  any  cubic  equation,  and  suppose  that  two  consecutive 
numbers  in  either  of  the  series 

1,  2,  3,  4,  &c. 
10,  20,  30,  40,  &c. 


0.1,  0.2,  0.3,  0.4,  &c. 
0.01,  0.02,  0.03,  0.04,  &c. 
&c.  &c. 


(A) 


are  found  such,  that  by  substituting  the  first  for  x  in  equa- 
tion (1),  the  result  shall  be  less  than  ./?4,and  by  substituting 
the  second,  the  result  shall  be  greater  than  Ji^  ;  then,  by  Art. 
263,  the  fust  of  these  numbers,  omitting  the  cyphers  if  it 
have  any,  will  be  the  first  figure  of  one  of  the  roots.  Let 
this  figure  be  denoted  by  n,  and  the  other  succeeding  figures 
of  the  same  root  by  r?,  7-3,  r4,  &c.,  respectively.  That  is, 
7-1,  ra,  ra,  Ta-,  &c.,  are  the  local  values  of  the  successive  fig- 
ures of  the  root. 

If  for  X,  in  equation  (1),  we  substitute  its  first  figure  n, 
we  shall  have  Air\-\-Air\  +^3ri  =  ^4.  (2) 

a 

Therefore,         r,  =  .5-^^-3^  (3) 


NUMEKICAL  SOLLTION  OF  IIIGHEU  EQUATIONS.  393 

If  we  put  3/  for  the  excess  of  the  true  root  above  its  first 
figure,  we  shall  have  a- ^  ri-f-y  ;  this  value  being  substi- 
tuted in  (1),  we  get 

A,f-{-  A'.y'^  +  A'.,y   -^   B    =   A„ 

or  A,y'-\-A'2f-\-A',y  =  A'„  (4) 

where  A'  ,  =  A+^An,  (1)  ) 

^';,  =  ^3-f-2^erj  +  3Ar^,  (2)  [  (B) 

Equation  (4)  is  in  all  respects  similar  to  the  original  equa- 
tion (1);  therefore,  repeating  the  above  process  upon  this 
equation,  we  shall  obtain 

where  ro  is  the  first  figure  of  the  root  of  equation  (4),  or  the 
second  figure  of  the  root  of  equation  (1).  Putting  z  for  the 
sum  of  all  the  remaining  figures,  we  have  y  =?-^,-|-z;  this 
value  substituted  in  (4),  gives 

A',z-\-A',r^_=A'^, 

A'zz''-{-2A'.r,z-\-AUrl  =  A'^y^, 

A^z'-\-3AirsZ'-\-  3Airlz-f-  Air^=  A^f, 

A,z^-^  A",z'-\-    A",z-^    B'    =   A\, 

or  A^z'-\-A",z--}-A"3Z  =  A"4,  (6) 

where     ^",  =  j^'a-fS^.r.,,  (1) 


A",  =  A'3-\-2A',r-2-^3A,rl,  (2)}    (B') 

A^"  =  A\  —  A',ro  —  A'2rl  —A,rl  (3) 

Here,  again,  equation  (6)  is  similar  to  equations  (4)  and  (1) 
50 


394        NUMERICAL    SOLUTION"    OF    HIGHER    EQUATIONS. 

We  might  now  proceed  to  find  the  first  figure  of  the 
root  of  equation  (6),  the  value  of  which  must  be  such,  that 
we  shall  have 

(306.)  Now,  by  observing  the  formation  of  the  coeffi- 
cients ./3'j,  .^'2,  in  equation  (5),  and  recollecting  that  7-1 
being  the  first  figure  of  the  root,  must  be  greater  than  rg,  it 
will  appear  obvious  that  ^^'y  must  constitute  the  largest 
portion  of  ^'3  + J^'grg-f- .iir^,  which  is  the  denominator 
of  the  value  u  as  given  by  (5),  and  if  nis  already  known, 
then  (2),  of  (B),  will  make  known  J^'g,  which  maybe  used 
as  a  trial  divisor  for  finding  ro,  the  second  figure  of  the 
root ;  the  same  may  be  observed  of  the  succeeding  divisors, 
and  it  is  obvious  that  these  trial  divisors  j3"3,  ^'"3,  &c., 
will  continually  approach  nearer  the  true  divisors. 

(307.)  If  we  multiply  the  first  coefficient  by  r,,  and  add 
the  product  to  the  second  coefficient,  we  shall  find 

A<i+Air..  (8) 

Multiplying  expression  (8)  by  r,,  and  adding  the  product 
to  the  third  coefficient,  we  have 

^3H-Avr,  +  ^ir?.  (9) 

Multiplying  expression  (9)  by  rj,  and  subtracting  the 
product  from  A\^  we  have 

^.1  — -^ari  — ^2^7  —  ArJ.  (10) 

Again,  multiplying  the  first  coefficient  by  ri,  and  adding 
the  product  to  expression  (8),  we  have 

^2+2^ir,.  (11) 

Multiplying  expression  (11)  by  7-j,  and  adding  the  pro- 
duct to  expression  (9),  we  have 

^3+2Ari  +  3^ir?.  (12) 


NUMERICAL    SOLUTION    OF    HIGHER    EQUATIONS.        395 

Again,  multiplying  the  first  coefficient  by  ?•  ,  and  adding 
the  product  to  expression  (11),  we  have 

A+3^iri.  (13) 

Expressions  (13),  (12),  and  (10),  are  the  values  of  the 
coefficients  ^5'o,  ^^'j  and  .^'i  respectively,  of  equation  (4), 
as  given  by  (B). 

(308.)  From  the  above  method  of  operation,  we  dis- 
cover that  the  root  of  a  cubic  equation  having  numerical 
coefficients,  can  be  found  by  the  following 

RULE. 

I.  Having  found  the  first  figure  of  the  root,  multijdy  it 
into  the  first  coefficient^  and  add  the  iproduct  to  the  second 
coefficient^  which  sum,  multiply  by  the  same  figure  and  add 
the  product  to  the  third  coefficient^  imiltiply  this  last  sum  by 
the  same  figure  and  subtract  the  product  from  the  term 
which  constitutes  the  right-hand  member  of  the  equation; 
the  remainder  we  shall  call  the  first  dividend. 

Again,  multiply  the  first  coefficient  by  the  same  figure, 
and  add  the  product  to  the  lust  number  under  the  second 
coefficient,  which  sum  must  be  multiplied  by  the  same  figure 
and  the  product  added  to  the  last  number  under  the  third 
coefficient,  the  result  we  shall  call  the  first  trial  divisor. 

Again,  multiply  the  first  coefficient  by  this  same  figure, 
and  add  the  product  to  the  last  number  under  the  second 
coefficient. 

II.  Find  the  second  figure  of  the  root  by  dividing  the 
FIRST  dividend  by  the  first  trial  divisor,  proceed  with 
this  second  figure  precisely  as  was  donewiththe  first  figure, 
observing  to  keep  the  work  so  that  units  shall  stand  under 
units,  tens  under  tens,  ^c,  ^c. 

Note. — The  above  rule  bears  a  close  resemblance  to 
the  rule  for  extracting  the  cube  root  of  a  polynomial,  as 
given  under  Art.  106. 


39f5 


NUMERICAL  SOLUTION  OF  HIGHER  EQUATIONS. 


EXAMPLES. 

1.  Find  a  root  of  the  cubic  equation  3a:'-j-2x'-j~'^^^='^^- 


OPERATION. 


<U    4)    fc.    H 


3     2 


14 

20 

215 

230 

245 

^171 

2492 

2513 

25151 

25172 

25193 

251957 


4  75(2.57  7  9  &c.=  a;. 

20  40 

48  =:  Ist  trial  divisor.  — 

5875  35  =  first  dividend. 

7025  =  2d  trial  divisor.        29375 

719797  

73724 1  =  3d  trial  divisor.       5625  =  2d  dividend. 
73900157  503S579 

74076361  =4th  trial  divisor.  

7409903713  586421 =  3d  dividend. 

517301099 


691 19901  =4th  dividend. 
66689133417 


2430767583. 


Find  one  of  the  roots  of  the  equation 
lx--^x^  —  Ux=lG15. 


1 

43 

85 
127 
1284 
1298 
1312 
13162 
13204 
13246 
132509 
132558 
132607 
1326112 


14 
244 

754 

77968 

80564 

8135372 

8214596 

822387163 

823315069 

82339463572 


1675(6.2676&c.=  x. 

1464 

211 
155936 


55064 
48812232 

6251768 
5756710141 

495057859 
4940367S1432 

1021077568. 


NUMERICAL  SOLUTION  CU  HIGHER  EQUATIONS. 


39-: 


Find  a  i 

oot  of  the  equation 

Zx' 

-2a:--f-a-=3. 

—  2 

1 

3(1.1417&c.=:x 

1 

2 

2 

4 

6 

— 

7 

673 

1 

73 

749 

673 

76 

78108 

79 

81364 

327 

802 

S144663 

312432 

814 
826 

SI 52929 

815871877 

1456S 

8263 

8144663 

8266 

8269 

6423337 

82711 

5711103139 

712233861. 

(309.)  The  above  roots  are  all  positive.  We  will  now 
give  a  couple  of  examples  when  the  value  of  a:  is  negative. 
The  operation  will  remain  the  same  if  we  observe  the  alge- 
braic rule  for  the  signs. 

4.   Find  a  root  of  the  equation  5x^  —  6x--f-3.r=  —  85. 


—  6 

3 

—  86(— 2.16139&C. 

—  16 

35 

—  70 

—  26 

87 

—  3'; 

9065 

—  15 

—  .365 

9435 

—  9065 

-370 
-375 

966180 
989040 

-5935 

—  3780 

98942405 

—  5797080 

—  3810 

—  3840 

989S0SI5 
9S99233995 

—  137920 

—  38405 

9900386535 

—  98942405 

-  38410 

—  38415 

990073231455 

—  38977.595 

-384165 

—  29697701985 

—  384180 

—  384195 

-9279893015 

-3841995 

—  8910659083095 

36923393J905, 


398  NUMERICAL  SOLUTION  OF  HIGHER   EQUATIONS. 

5.  Find  a  root  of  the  equation  x^-{-x'^-{-'70x=^ — 300. 


1 

70 

—  300(— 3.73S79&C 

—2 

76 

—  228 

—5 

91 

—8 

9709 

—  72 

—87 

10367 

-67963 

—94 

—  101 

1039739 
1042787 

—  4037 

—  1013 

1043602S4 

—  3119217 

—  1016 

—  1019 

104441932 
10444908229 

—  917783 

—  10198 

10445623307 

—  834882272 

— 10206 
—  10214 

1 04457 1525''71 

IVIIC/I  lKj^%J^t     I 

—  82900728 

—  102147 

—  73114357603 

—  102154 

—  102161 

—  9786370397 

—  1021619 

—  9401143727439 

.385226600561. 


(310.)  When  the  second  or  first  power  of  a:  is  wanting, 
we  must  consider  its  coefficient  as  being  d=0. 

6.  Find  a  root  of  the  equation  x^ — ]2x  =  28. 


±0 

—  12 

28(4.30213&c.  =  a;. 

4 

4 

16 

8 

36 

— 

12 

3969, 

12 

123 

4347 

11907 

126 

43495804 

129 

43521612 

93 

12902 

4352290261 

86991608 

12904 
12906 

4352419323 
435245804199 

6008392 

129061 

4352290261 

129062 



129063 

1656101739 

1290633 

1305737412597 

350364326403. 


NUMERICAL  SOLUTION  OF  HIGHE-R    EQUATIONS. 


599 


7.  Find  a  root  of  the  equation  2x^-\-3x^  =  850. 


3 

17 
31 

±0 
119 
336 

850(7.05025&c.=  x 
833 

45 

3382550 

17 

4510 
4520 

3405150 
34052406008 

16912750 

4530 

34053312024 

87250 

453004 
453008 

3405353853050 

6S104812016 

453012 

19145187984 

4530130 

17026769265250 

211841"<718750. 

(311.)  In  all  the  preceding  examples,  the  first  figure  of 
the  root  has  been  in  the  units'  place  ;  \\c  will  now  add  two 
examples  in  which  the  first  figure  is  in  the  tens'  place. 

8.  Find  a  root  of  the  equation  3a:^  — 7a;'' -|-I3x  =  45000. 


-7 

13 

45000(25.404&c.=:x 

53 

1073 

21460 

113 

3333 

173 

4273 

23540 

188 

■5288 

21365 

203 

537568 

218 

5-l63«4 

2175 

2192 

5464726448 

2150272 

2204 

2216 

24728 

221612 

21S58905792 
286909420?, 

"ind  a 

•oot  of  the  equation 

.rS^GOx-  — 800x^360000 

60 

-800 

60000(30.537&c.=z 

90 

1900 

57000 

120 

5500 



150 

:,57525 

3000 

1505 

565075 

27876-23 

1510 

56552959 

1515 

56598427 

212375 

15153 

5660903S79 

160n5S877 

15156 
15159 

42716123 

151597 

39626327153 
30^979.^847. 

400       NUMERICAL    SOLUTION    OF    HIGHER    EQUATIONS. 

(312.)  The  method  of  proceeding,  "when  the  first  figure 
is  of  a  local  value  greater  than  ten,  will  be  obvious. 

We  will  add  two  examples  in  which  the  fii;st  figure  is  in 
the  tenth's  place. 

10.  Find  a  root  of  lOz^  _  24x^  —  30x  =  —  6. 
10         —24  —30  —6  (0.1768 &c.=x. 

—  23  —323  —323 


-21 

—  35921 

—  277 

-203 

—  37293 

—  251447 

-196 

—  189 

—  3740604 
-3751872 

—  25553 

-1884 

—  275336896 

—  22443624 

-1878 
—  1872 

—  3109376 

—  18712 

—  3002695168 

—  106680832. 

Find  a  root  of  the  equation  x^-\-  9a;  =  6. 

iO 

9 

6  (0.6378  &c.=a: 

0.6 

936 

5616 

12 

1008 

18 

101349 

384 

183 

101907 

304047 

186 

10203979 

189 

10217307 

79953 

1897 
1904 

1021883644 

71427853 

1911 

8525147 

19118 

8175069152 

350077848. 
(313.)  By  reviewing  the  foregoing  examples,  we  discover 
that  the  last  terms  under  tlie  third  coefficient,  or  the  trial 
divisors^  remain  unchanged  in  several  of  its  kft-hand  fig- 
ures, thus  in  example  1,  740  is  common  to  the  left-hand  of 
the  last  two  terms  under  the  third  coefficient.  In  example 
2,  the  figures  common  are  8233.  In  example  3,  the  figures 
common,  are  815,  and  so  for  the  succeeding  examples. 


NUMERICAL  SOLUTION  OF  HIGHER   IQUATIONS. 


401 


the  constant  figures  of  the  last  term  under  the  third  co- 
efficient, and  also  omit  from  the  right  of  the  last  dividend, 
the  same  number  of  figures  save  one,  we  may  then  divide  the 
remaining  figures  of  the  dividend  by  the  remaining  figures 
of  the  last  term  under  the  third  coefficient,  by  long  division  ; 
and  as  many  additional  figures  of  the  root  may  in  this  way 
be  found  as  there  are  figures  in  the  divisor  thus  used. 

12..  Find  a  root,  to  8  decimals,  of  the  equation 
x3+x-  =  500. 

500(7. 61727975&c.=x. 


1 

8 
15 
22 
226 
232 
238 
2381 
2382 
2383 
23837 
23844 
23851 
238512 
258514 


iO 

56 
161 
17456 
18848 
1887181 
1889563 
189123159 
189290067 
18929483724 
18929.960752 


108 
104736 

3264 

1887181 

1376819 
1323862113 

52956887 
37858967448 

150979.19552 
132509 

18470 
17036 

1434 
1325 


109 
95 


61 


402 


NUMERICAL  SOLUTION  OF  HIGHER  EQUATIONS. 


EXPLANATION. 

After  tinding  4  decimal  places  in  the  root  by  the  prece- 
ding rule,  we  cut  off  6  figures  from  the  right  of  the  last  trial 
divisor,  thus  leaving  the  constant  figures  18929 ;  and  from  the 
right  of  the  dividend  we  cut  off  5  figures,  leaving  150979  ; 
we  then  divided  150979  by  18929,  by  the  rule  for  abridged 
division,  (see  Higher  Arithmetic,)  and  thus  obtained  the 
additional  figures  of  the  root. 

13.  Find  a  root,  to  10  decimals,  of  the  equation 
r^  — 17a;2-|-54x  =  350. 


17 

54 

350)14.9540686096. 

-7 

—  16 

—  160 

3 

14 

13 

82 

510 

17 

166 

328 

21 

18931 



25 

21343 

182 

259 

2148175 

70379 

268 

2162075 

277 

2163189 

11621 

2775 

216430348 

10740875 

2780 

2164320197236 

2785 

880125 

27854 

865275664 

27858 

27862 

14849336 

2786206 

12985921183416 

186341.4816584 

173147 

13194 

12986 

208 

194 

14 

13 

NUMERICAL  SOLUTION  OF  HIGHER  EQUATIONS. 


403 


314.)  Thus  far  we  have  sought  only  one  of  the  roots  of 
v)ur  equations.  If  we  wish  the  three  roots,  we  may  divide 
the  given  equation,  when  all  the  terms  are  transposed  to  one 
side,  by  the  unknown  quantity  ^niyius  the  value  of  the  root 
found  by  the  above  method,  we  shall  thus  depress  the  cubic 
equation  to  a  quadratic.     (See  Art.  255.) 

14.  Find  the  three  roots  of  the  equation 

Here,  we  soon  discover  that  one  of  the  roots  lies  between 
1  and  2  ;  seeking  this  root  by  the  above  process,  we  find 


-15 

63 

50(1.028039231  &c.= 

—  14 

49 

49 

—  13 

36 

— 

—  12 

357604 

I 

—  1198 

355212 

715208 

1196 

35425744 

—  1194 

35330352 

284792 

—  11932 

353299945209. 

283405952 

—  11924 

—  11916 

35329  6370427 

^o*j^j»\JOt  yj^jirl 

1386048 

—  1191597 

• 

1059899835627 

—  1191594 

326148.164373 
317967 


8181 
7066 


1115 
1060 


Now,  dividing  x^  —  15x=  +  63x  —  50  by  x—1 .02803923 
we  find,  for  a  quotient,  the  following  : 


404       XIMEKICAL    SOLUTION    OF    HIGHER   EQUATIONS. 

>r'^— 13.97 196077X+48.6362762. 
Hence,  we  have  this  quadratic  equation, 

x3—  13.97196077a:  =  —  48.6362762. 

This  solved  by  the  usual  rule  for  quadratics,  gives  the 
following  values  : 

x  =  6.576535;  a:  =  7.395426. 
Therefore,  the  three  roots  of  x^ —  15a;"--|-63a:  =  505  are 
1.028039  ;  6.576535  :  7.395426. 

(315.)  From  the  work  of  the  last  example,  we  see  that 
we  need  only  seek  one  of  the  roots  of  a  cubic  equation  by 
the  foregoing  rule,  as  the  other  two  may  then  be  found  by 
the  solution  of  a  quadratic.  When  all  the  three  roots  are 
real,  it  will  frequently  be  as  simple  to  find  them  by  the 
foregoing  general  method.  But  when  two  of  the  roots  are 
imaginary,  we  must  proceed  agreeably  to  the  last  Art. 

15.  Find  the  three  roots  of  the  equation  t'* — 15x= — 21. 

Applying  the  principle  §f  Sturm's  Theorem,  we  find^ 

X  =x^—\5x-\-2\, 
Zi  =  a:^  — 5, 
X,  =  lOx  — 21, 
^3  =  59. 

For  X  =  —  00,  we  find j h  =  ^  variations. 

"    x=H-oo,       "       -4-  -I-  -j--^=0         « 

Therefore,  this  equation  has  three  real  roots. 


""or  X  = 

—  5,  we  fir 

id 1 p  =3  va 

riat 

"    a:=. 

-4,       " 

+  +  -  +  =2 

u 

«   x  = 

1,       " 

+ +=2 

(( 

"   x  = 

2,       " 

+  =1 

u 

«   x  = 

3,      " 

-f--f4-+=o 

a 

NUMERICAL    SOLITIOX    OF    HIGHER    EQUATIONS,        40J 

Hence,  there  is  one  negative  root  between  — 4  and — 5  ; 
one  positive  root  between  1  and  2  j  and  one  positive  root 
between  2  and  3. 

For  the  positive  root  between  1  and  2,  we  have  the  fol- 
lowing 


OPERATION. 

±0 

—  15 

—  21(1.769149632  &c.=x 

1 

—  14 

—  14 

2 

—  12 

3 

—  941 

-7 

37 

—  633 

—  6587 

44 

—  60204 

51 

—  57072 

—  413 

516 

—  5659599 

—  361224 

522 

—  5611917 



528 

—  561138629 

—  51776 

5289 

—  561085557 

—  50936391 

5298 

—  5610.6432764 

5307 

—  839609 

53071 

—  561138629 

53072 

53073 

—  278470371 

530734 

—  224425731056 

—  54044.639944 

-  50495 

—  3549 

—  3366 

—  183 

-168 

—  15 

—  11 

406        NUMERICAL    SOLUTION    OF    HIGHER    EQUATIONS 


For  the  negative  root,  we  have  the  following 


OPERATION. 

lO 

-15 

-21(— 4.4416216£ 

-4 

1 

—  4 

-8 

33 

12 

3796 

—  17 

124 

4308 

—  15184 

128 

436096 

132 

441408 

-1816 

1324 

44154121 

—  1744384 

1328 
1332 

44167443 
4417543716 

—  71616 

13321 

4418343168 

—  44154121 

13322 
13323 

4418.36981764 

—  27461879 

133236 
133242 

—  26505262296 

133248 

—  956616704 

1332482 

—  883673963528 

—  7294.2740472 

—  4418 

—  2876 

—  2651 

—  225 

—  221 

—  4 

—  4 

NUMERICAL    SOLUTION    OF    HIGHER    EQUATIONS.        40' 

For  the  positive  root  between  2  and  3,  we  have  this 


OPERATION. 

±0 

—15 

-21(2.67247201 

2 
4 
6 

-11 

—  3 

96 

—  22 

1 

66 

528 

576 

72 

58309 



78 

63867 

424 

787 

6402724 

408163 

794 

6418752 

801 

642195856 

15837 

8012 

642516528 

12805448 

8014 

6425.7264889 

8016 

3041552 

80164 

2568783424 

80168 

80172 

462768576 

801727 

449800854223 

12967.721777 

12851 

116 


Hence,  the  three  roots  of  x^ —  15a;  = — 21,  are 
x  =  — 4.441621651  ;  1.769149632;  2.672472018. 
16.  Find  the  three  roots  of  the  equation 

lOOOOr^  —  4519x2-f-  665a:  =  32. 
Applying  Sturm's  Theorem  to  tiiis  equation,  we  find 
X  =  10000x3  — 4519x2+665x— 32, 
Xi  =  BOOOOx-'—  9038x-f665, 
X2=942722x— 125135, 
X3  =  5425404570000. 


408  NUMERICAL  SOLUTION  .OF  HIGHER  EQUATIONS. 


When  a:  ^  0,  w'^  find ] h  =  ^  variations, 

«         X=l,  "  +4-    +   +::=.0  " 

Hence,  the  equation  has  three  positive  roots,  each  less 
than  1. 

When  X  =0,1,  we  find 1 h  =  ^  variations, 

"      a:=0.2,       "       +  +  -f+  =  0  " 

Which  shows  that  the  first  figure  of  each  root  is  0.1. 

Again,  when  x  =  0.11,  we  find 1 [-=  3  variations. 

"     a:  =  0.12,      "      -i--\ \-=2 

+ +  =2 

+  +  =  1 

a:  =  0.19,      "      — +  +  -}-  =  l 
From  this,  we  discover  that  the  first  two  figures  of  the 
least  root  are  0.11 ;  the  first  two  figures  of  the  next  root  are 
0.13 ;  the  first  two  figures  of  the  other  root  are  0.19. 
For  the  first  root  we  have  the  following  : 


"      a;  =  0.13, 
"     x  =  0.14. 


10000     -4519 

665 

32  (0.119503816&C. 

—  3519 

3131 

3131 

—  2519 

612 

-1519 

4701 

69 

—  1419 

3382 

4701 

-1319 

23659 

—  1219 

14308 

2199 

—  1129 

138360 

212931 

—  1039 

133665 

—  949 

1336.369809 

6969 

—  944 

6918 

—939 

—  934 

61 

—93397 

400910942V 

10908.90573 

10691 

217 

134 

.. 

83 

80 

m^ 

1. 

NUMERICAL  SOLUTION  OF  HIGHER  EQUATIONS.         409 

For  the  second  root  we  have  the  following 
10000 


OPERATION. 

-4519 

665 

32  (0.137139216&C 

—  3519 

3131 

3131 

—  2519 

612 

—  1519 

2463 

69 

-1219 

—  294 

7389 

—  919 

—  6783 

—  619 

—  10136 

—  489 

—  549 

— 101768 

—  47481 

—  479 

— 102175 
— 10229671 

—  409 

—  1419 

—408 

— 10241833 

- 101768 

—  407 

— 1024.547809 

. 

—  406 

—  40132 

—  4057 

—  30689013 

—  4054 

—  4051 

—  9442987 

—  4030] 

—  9220930281 

—  2220.56719 

—  2049 

—  171 

—  102 

—  61 


48 


410       NUMERICAL    SOLUTION    OF   HIGHEB    EQUATIONS. 

For  the  third  root  we  have  the  following 
10000 


-4519 

665 

32  (0.195256967  &c 

—  3519 

3131 

3131 

—  2519 

612 

—  1519 

549 

69 

-619 

3078 

4941 

281 

36935 

1181 

43340 

1959 

1231 

436066 

184675 

1281 

438736 

1331 

43940475 

11225 

1333 

44007375 

872132 

1335 
1337 

440.1540636 

25036S 

13375 

219702375 

13380 
13383 

30665625 

133856 

26409243816 

4256.381184 
3962 


294 


Hence,  the  three  roots  of  the  equation 

10000x3  —  45 1 9x2+6650:  =  32, 
are  0.119503816;  0.137139216;  0.195256967. 

17.  Find  the  three  roots  of  the  equation  x^-{-2x- —  3x=9 

Applying  to  this  equation  the  Theorem  of  Sturm,  we  find 

X  =x'-|-2x'— 3a:— 9, 
Xi=  3x3-|-4x  — 3, 
X2  =  26x-j-75, 
X3  =  —  7047. 


NUMERICAL    SOLUTION    OF    HIGHER    EQUATIONS.        411 

When  X  =  —  oo,  we  find 1 =  2  variations, 

"      x  =  +oo,       "       +4-+_^l         " 
"      a:=  1,  "      —  -f   +— =  2         " 

"      X  =  2,  "      4-  +  -I-  —  =  1         " 

Then    this   equation    has   but  one  real  root,  which  lies 
between  1  and  2,  the  other  roots  being  imaginary. 
We  find  the  real  root  by  the  following  : 

1 


2 

3 

9(1.939465  &c. 

=  X. 

3 

0 

0 

4 

4 

_ 

5 

931 

9 

59 

1543 

8379 

68 

156619 



77 

158947 

621 

773 

15964891 

469857 

776 
779 

16035163 
1603.828996 

151143 

7799 

143684019 

7808 
7817 

7458981 

78174 

6415315984 

10436.65016 
9623 


813 
801 


Dividing  r^+2x2  —  3a:  —  9  by  a:- 
T^  +  3.939465X  -f  4.640455  for  the  quotient. 
Therefore,  solving  the  quadratic 

x2-f3.939465x  =  — 4.640455, 
we  find  the  following  imaginary  roots, 

1.96973  +  0.87213  v/^, 
.96973  — 0.87213  v^—1. 


(-1.9 
^-1.9 


412         NUMERICAL  SOLUTION  OF  HIGHER  EQUATIONS. 

18.  Find  the  three  roots  of  the  equation  x^ —  5x^-\-8x=l. 
By  Sturm's  Theorem  we  have  already  found,  page  381, 


X  =x3  — 5a;2+8x— 1, 
Xt  =  Zx''—10x-{-S, 
Xo  =  2x  — 31, 
X3  =  — 2295. 

When  x  =  —  oo  ,  we  find 1 :=  2  variations, 

+  +  +-  =  1         " 
—  H =2     '    " 


(( 

X=:  +  CO, 

u 

u 

x  =  0, 

C( 

11 

.r  =  l. 

u 

+  + 


=  1 


Hence,  this  equation  has  one  positive  root  which  lies  be- 
tween 0  and  1,  and  two  imaginary  roots. 
Its  real  root  is  found  by  the  following 


1      — 


OPERATION. 

5 

8 

1  (0.1362934&C 

49 

751 

751 

48 

703 

— 

47 

68899 

249 

467 

67507 

206697 

464 

6723076 

461 

6695488 

42303 

4604 

669.456964 

40338456 

4598 

4592 

1964544 

45918 

1338913928 

6256.30072 


231 
201 

30 
27 


By  dividing    x''  —  5x'+8x— 1  by  x  — 0.1362934, 


NUMERICAL  SOLUTION  Or  IIIGHEU  EQUATIONS,         413 

we  find  the  quadratic  x^  — 4.8637066a:  =  —  7.3371089, 

which  gives  the  following  imaginary  roots : 
2.43185  +  1. 19298  v^^, 
2.43185  — 1.19298  v/—l. 


x  =  - 


19.  Find  one  of  the  roots  of  x^  —  •2x  =  5. 

Ans.  a:  =2.09455 148&C. 

20.  Find  one  of  the  roots  of  2x^-}-3x  =  90. 

^  Ans.  a;=3.41639726&c. 

21.  Find  one  of  the  roots  of  x^-\-X'-\-x  =  100. 

Ans.  a:  =  4.26442997&c. 

22.  Find  one  of  the  roots  of  x^-|-x=:  500. 

Ans.  a;  =  7.89500828&c. 

23.  Find  one  of  the  roots  of  a:^+10x'+5x  =  2600. 

Ans.   11.00679933&C. 

SOLUTION  OF  EQUATIONS  ABOVE  THE  THIRD  DEGREE. 

(316.)  It  is  obvious  that  the  above  method  which  we  have 
employed  for  cubic  equations,  will  apply  equally  well  to 
equations  of  a  superior  degree.  By  carefully  studying  the 
preceding  method,  we  shall  be  able  to  deduce,  for  equations 
of  the  nth  degree,  this  general 

RULE. 

1.  Having  found  the  first  figure  of  the  root,  multiply  it 
into  the  first  coefficient  and  add  the  product  to  the  second 
coefficient,  which  sum  multiply  by  the  same  figure  and  add 
the  product  to  the  third  coefficient,  and  this  sum  must  be 
multiplied  by  the  same  figure  and  the  product  added  to  the 
fourth  coefficient;  and  so  continue  to  multiply  the  last  re- 
sult by  this  same  figure  and  to  add  the  product  to  the  next 


414        NUMERICAL    SOLUTION    OF    HIGHER    EQUATIONS. 

succeeding  coefficient^  until  the  last  coefficient  is  reached^ 
which  last  sum  must  he  multiplied  by  the  same  figure  and 
the  product  subtracted  from  the  term  constituting  the  right- 
hand  member  of  the  equation ;  the  remainder  we  will  call 

the  FIRST    DIVIDEND. 

Jigain,  multiply  the  first  coefficient  by  the  same  figure^ 
and  add  the  product  to  the  number  under  the  second  coeffi- 
cie7it,  which  sum  must  be  multiplied  by  the  same  figure^ 
and  the  product  added  to  the  term  under  the  third  coefficient ; 
and  thus  we  must  continue  to  multiply  and  add,  until  we 
have  obtained  the  second  term  under  the  last  coefficient,  winch 
result  we  shall  call  the  first  trial  divisor. 

Again,  multiply  the  first  coefficient  by  the  same  figure  of 
the  root,  and  add  the  product  to  the  last  term  under  the 
second  coefficient,  which  result  must  be  multiplied  by  the 
same  figure,  and  the  product  added  to  the  last  number  under 
the  third  coefficient;  and  thus  we  must  continue  to  multiply 
and  add  until  we  reach  the  coefficient  next  to  the  last,  wheji 
we  must  again  begin  with  the  first  coefficient  and  multiply 
and  add  as  before,  until  we  reach  the  n  — 2th  coefficient; 
then,  again,  commencing  with  the  first  coefficient,  we  must 
multiply  and  add  until  we  reach  the  n  —  3rf  coefficient;  ive 
must  continue  this  process,  until  we  have  thus  obtained  n 
terms  under  the  second  coefficient,  n  —  1  terms  under  the 
third  coefficient,  n  —  2  terms  under  the  fourth  coefficient, 
n  —  3  terms  under  the  fifth  coefficient,  and  so  of  the  rest. 

II.  Find  the  second  figure  of  the  root,  by  dividing  the 
FIRST  dividend  by  the  first  trial  divisor,  proceed  with 
this  second  figure,  precisely  as  was  done  with  the  first  figure, 
observing  to  keep  the  work  so  that  units  shall  stand  under 
units,  tens  under  tens,  Sfc,  S)C. 

EXAMPLES. 

1.  Find  one  of  the  roots  of  the  equation 
3x^_|.x3-f  4a:-'-f  5x  =  375. 


NUMERICAL   SOLUTION   OF   HIGHER   EQUATIONS.        415 


1 
10 
19 
28 
37 
373 
376 
379 
382 
3829 
3838 
3847 
3856 
38569 


4 

34 

91 
175 

17873 

17249 

18628 

1874287 

1885801 

1897342 

189849907 


OPERATION. 
5 
107 
380 
397873 
416122 
421744861 
427402264 
427.971813721 


375(3.13364  &c.=i. 

321 


156035417 
1283915441 163 


2764.387288^7 
2567 


2.  Find  the  four  roots  of  the  equation 
X*  —  80x3  -|_  l99Sa;2  _  14937a;  = 


197 
171 


—  5000. 


1    — 


—  80 

—  50 

—  20 
10 
40 
44 
48 
52 
56 
668 
576 
684 
592 
5923 
5926 

1998 
498 
-102 
198 
374 
566 
774 
81944 
86552 
91224 
9140169 
9157947 
9175734 
917742041 
917860692 

FIRST    OPERATION. 
—  14937 
3 

-  5000(34.83228  &c.  =  x 
90 

—  1561                  - 
703                - 

-5090 
6244 

i050968 
■2G783.SS507 
210586234S 
21076978320S8 
"  210.9533553472 

1154 

10868416 

671584 
6235165521 

480674479 
4215395664176 

591.349125824 


168 
t. 


416         NUMERICAL  SOLUTION  OF  HIGHER  EQUATIONS. 


SECOND   OPERATION. 


1     — 


—80 

1998 

— 14937 

—  5000(32.06029&c 

-60 

498 

3 

90 

—20 

—  102 

-3057 

10 

198 

-2493 

—  5090 

40 

282 

-1753 

—  4986 

42 

370 

—  1725106984 

44 

462 

—  169.7040736 

—  104 

46 

4648836 

— 10350641904 

48 

4806 

4677708 

—  49.358096 

4812 

—  34 

—  15 

—  15 

THIRD   OPERATION. 


-80 

1998 

_ 14937 

—  5000(1 2.75644&C 

—70 

1298 

—  1967 

— 19570 

—  60 
—CO 

698 
198 

5023 
5267 

14570 

—40 

122 

5367 

10534 

—  88 

50 

5339063 

-36 

-18 

5296132 

4036 

—  34 

—  3991 

5291946125 

37373441 

-32 
-313 

—  6133 

—  8226 

5287.687500 

2986559 

—  306 

—  837175 

26459730626 

—  299 

—  292 

—  861726 

34058.59375 

—  2915 

31726 

-2910 

2332 
2115 


217 
212 


NUMERICAL   SOLUTION   OF  HIGHER    El^UATlONS 


AV 


FOURTH     OPEUAiTON. 


80 

1993 

—  14937 

—  5000         (0.3509SSCC 

797 

197409 

—  14344773 

—  '13034319 

794 

195027 

13759C92 

791 

192C54 

—  13663561875 

— 69656S1 

788 

19226025 

— 135.G763S500 

—  68317809375 

7875 

19186675 

7870 

—  133.9000625 

122 


-11 

—  10 


Hence,  the  four  roots,  true  to  5  decimal  places,  are 

34.83228;  32.06029;  12.75644;  0.35098. 

3.  Find  the  four  roots  of  the  equation 

a;i_j_4x '  -  4a-  —  1  la;+4  =  0. 

By  Sturm's  Theorem  ^ve  have 

X  =a;'-|-4.r<  — 4j:-— lLi-f-4:=0, 
X,  =  4x3 -j-  i2x^  _Sx—U, 


X, 

==20x 

-4-25X-  — 27, 

X,- 

=  227x-|-31, 

X,-. 

=  154' 

798S 

riicn  x  =  - 

—  5,  we  fin 

d  +  -  H 1-=.4  V 

ariations 

"       x  =  - 

-4, 

a 

-++-+-3 

a 

"       x  =  - 

_  o 

a 

h  +  -+  =  3 

a 

«      x  =  - 

-h 

ii 

+  +  --  +  -2 

" 

"      x  = 

0, 

u 

+ +  +--2 

a 

"      x  = 

1, 

u 

++  +  =  1 

u 

"       x  = 

2, 

u 

+  4-4-  +  +--0 

u 

Hence,  there  is  one  negative  root  between  — 5  and  -  4  ; 
one  negative  root  between  —2  and  —  1  ;  one  positive  root 
between  0  and  1  ;  and  one  positive  root  between  1  and  2. 


41S  NUMERICAL  SOLI  TION   OK   HIGHER   EQliATIONS. 

These  roots  wiien  found  are 

ar  =  — 4.2834, 
x  =  — 1.6908, 
x=  0.3373, 
x=      1.6369. 

4.   Find  one. of  the  roots  of  the  equation 

2x'-j-ox'-}-  6x-^+  2x'  —  3x  =  300. 


OPERATION. 

.) 

6 

2 

—  3 

300(2.2233498cc. 

9 

24 

50 

97 

194 

13 

50 

150 

397 

— 

17 

84 

318 

465S432 

106 

21 

120 

344210 

540131^0 

9310804 

95 

1310S 

37I4G4 

54819U13C32 

254 

13024 

399700 

5503O3425C0 

12SGI3G 

25S 

14148 

402700816 

5575300G592'i562 

109038027204 

202 

140S0 

405064464 

558.759248434410 

20 '• 

1473408 

409632960 

18075572736 

270 

147^«24 

409080108854 

1072591397779680 

2704 
2708 

1484248 
1489060 

409527502010 

1949.05275820214 

2712 

149019filS 

1676 

2716 

149131254 



2720 

273 

27200 

223 

27212 

— 

5.  Find  one  of  ihe  roots  of  the  equation 
2x5  — 7i3+10x  =  9 


NUMERICAL  SOLUTION  OF  HIGHER  EQUATIONS. 


419 


OPERATION. 

0    - 

-7 

0 

10 

9(1.630101025  &c. 

2    _ 

—  5 

—  5 

5 

5 

4     - 

-1 

-6 

-  1 

- 

6 

5 

-  1 

54992 

4 

8 

13 

10832 

217760 

329952 

10 

1972 

27128 

2326581362 

J12 

2716 

48320 

2479627610 

70048 

124 

3532 

49660454 

248015150553963002 

6979744' 'Se 

136 

4120 

51015416 

24806.7544722052010 

148 

4.16S1S 

52384940 

25055914 

160 

451654 

52389553963002 

248015150553963002 

1606 
1612 

45650S 
4613S0 

;iO'-fo  1 1  fitin>iQnr»>3 

O/oy-J  loSl 

2543.989446036998 

1618 

4613063002 

2481 

1624 

4614126006 

1630 

62 

163002 

50 

163004 

12 

12 

6.  Find  one  of  the  roots  of  the  equation 

x'  +  2x*  +  3x3  -^  ix'  +  5x  =  54321. 


2 
10 
18 
26 
34 
42 
424 
428 
432 
436 
440 
4401 
4402 


3 

83 
227 
435 
707 
72396 
74108 
75836 
77580 
7762101 
7766803 


OPERATION. 


4 
668 
2  84 
5964 
6253584 
6550016 
6853360 
6861122401 
6868889204 


5 

5349 
25221 
277224336 
303424400 
3041105122401 
304.7974011605 


54321  (8.4144  &c. 

42792 


11529 
1108897344 


44002656 
3041105122401 


1359.160477599 
1219 


140 
122 


18. 


420  NUMERICAl-  SOLUTION   OK  HIGHER  EC^L  ATIONS. 

7.  Find  a  root  of  the  equation 

26a.*  +  28la.'3  -  576a:2  ^  29Sx  :^  25. 

Ans.  a:  =0.77933994  &c. 

8.  Find  a  root  of  the  equation 

x^  —  5x'3 -f-5a:^  1. 

Ans.  X  =  0.20905692  &c. 

9.  Find  a  root  of  the  equation 

x^-}.2x^'  +  3x*  +  4x'  +  5a:2+6.r=  654321. 

Ans.  a:  =  8.95697957  &c. 

10.  Find  a  root  of  the  equation 

2x'^  —  6x«  —  5x^  +  20r»  -j-  2x^  —  18a,-  +  4x  =  4. 
Ans.  a;  =  2.62599736  &c. 


r 


14  DAY  USE 

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